DOKK Library

Structure and Interpretation of Computer Programs - Second Edition

Authors Gerald Jay Sussman Harold Abelson Julie Sussman

License CC-BY-SA-4.0

Structure and
 of Computer

  second edition
  Unofficial Texinfo Format 2.andresraba5.6

  Harold Abelson and
  Gerald Jay Sussman
  with Julie Sussman
  foreword by Alan J. Perlis
    ©1996 by e Massachuses Institute of Technology

    Structure and Interpretation of Computer Programs,
                       second edition

         Harold Abelson and Gerald Jay Sussman
       with Julie Sussman, foreword by Alan J. Perlis

      is work is licensed under a Creative Commons
      Aribution-ShareAlike 4.0 International License
    (  .). Based on a work at

                     e  Press
                Cambridge, Massachuses
                   London, England

              McGraw-Hill Book Company
            New York, St. Louis, San Francisco,
                   Montreal, Toronto

Unofficial Texinfo Format 2.andresraba5.6 (February 2, 2016),
       based on 2.neilvandyke4 (January 10, 2007).

Unofficial Texinfo Format                                               ix

Dedication                                                           xii

Foreword                                                             xiii

Preface to the Second Edition                                        xix

Preface to the First Edition                                         xxi

Anowledgments                                                   xxv

1   Building Abstractions with Procedures                              1
    1.1 e Elements of Programming . . . . . . . . . . . . . .         6
         1.1.1 Expressions . . . . . . . . . . . . . . . . . . . .     7
         1.1.2 Naming and the Environment . . . . . . . . . .         10
         1.1.3 Evaluating Combinations . . . . . . . . . . . .        12
         1.1.4 Compound Procedures . . . . . . . . . . . . . .        15
         1.1.5 e Substitution Model for Procedure Application        18
         1.1.6 Conditional Expressions and Predicates . . . .         22
         1.1.7 Example: Square Roots by Newton’s Method . .           28

          1.1.8 Procedures as Black-Box Abstractions        .   .   .   .   .    33
    1.2   Procedures and the Processes ey Generate .       .   .   .   .   .    40
          1.2.1 Linear Recursion and Iteration . . . .      .   .   .   .   .    41
          1.2.2 Tree Recursion . . . . . . . . . . . . .    .   .   .   .   .    47
          1.2.3 Orders of Growth . . . . . . . . . . . .    .   .   .   .   .    54
          1.2.4 Exponentiation . . . . . . . . . . . . .    .   .   .   .   .    57
          1.2.5 Greatest Common Divisors . . . . . .        .   .   .   .   .    62
          1.2.6 Example: Testing for Primality . . . .      .   .   .   .   .    65
    1.3   Formulating Abstractions
          with Higher-Order Procedures . . . . . . . . .    .   .   .   .   .    74
          1.3.1 Procedures as Arguments . . . . . . .       .   .   .   .   .    76
          1.3.2 Constructing Procedures Using lambda        .   .   .   .   .    83
          1.3.3 Procedures as General Methods . . . .       .   .   .   .   .    89
          1.3.4 Procedures as Returned Values . . . .       .   .   .   .   .    97

2   Building Abstractions with Data                                             107
    2.1 Introduction to Data Abstraction . . . . . . . .    . . . . .           112
         2.1.1 Example: Arithmetic Operations
                for Rational Numbers . . . . . . . . . .    .   .   .   .   .   113
         2.1.2 Abstraction Barriers . . . . . . . . . .     .   .   .   .   .   118
         2.1.3 What Is Meant by Data? . . . . . . . .       .   .   .   .   .   122
         2.1.4 Extended Exercise: Interval Arithmetic       .   .   .   .   .   126
    2.2 Hierarchical Data and the Closure Property . .      .   .   .   .   .   132
         2.2.1 Representing Sequences . . . . . . . .       .   .   .   .   .   134
         2.2.2 Hierarchical Structures . . . . . . . . .    .   .   .   .   .   147
         2.2.3 Sequences as Conventional Interfaces         .   .   .   .   .   154
         2.2.4 Example: A Picture Language . . . . .        .   .   .   .   .   172
    2.3 Symbolic Data . . . . . . . . . . . . . . . . . .   .   .   .   .   .   192
         2.3.1 otation . . . . . . . . . . . . . . . .     .   .   .   .   .   192

          2.3.2 Example: Symbolic Differentiation . . . . .          .   .   197
          2.3.3 Example: Representing Sets . . . . . . . . .        .   .   205
          2.3.4 Example: Huffman Encoding Trees . . . . .            .   .   218
    2.4   Multiple Representations for Abstract Data . . . . .      .   .   229
          2.4.1 Representations for Complex Numbers . . .           .   .   232
          2.4.2 Tagged data . . . . . . . . . . . . . . . . . .     .   .   237
          2.4.3 Data-Directed Programming and Additivity            .   .   242
    2.5   Systems with Generic Operations . . . . . . . . . .       .   .   254
          2.5.1 Generic Arithmetic Operations . . . . . . .         .   .   255
          2.5.2 Combining Data of Different Types . . . . .          .   .   262
          2.5.3 Example: Symbolic Algebra . . . . . . . . .         .   .   274

3   Modularity, Objects, and State                                          294
    3.1 Assignment and Local State . . . . . . . . . . .    .   .   .   .   296
        3.1.1 Local State Variables . . . . . . . . . . .   .   .   .   .   297
        3.1.2 e Benefits of Introducing Assignment          .   .   .   .   305
        3.1.3 e Costs of Introducing Assignment . .        .   .   .   .   311
    3.2 e Environment Model of Evaluation . . . . . .      .   .   .   .   320
        3.2.1 e Rules for Evaluation . . . . . . . . .     .   .   .   .   322
        3.2.2 Applying Simple Procedures . . . . . . .      .   .   .   .   327
        3.2.3 Frames as the Repository of Local State       .   .   .   .   330
        3.2.4 Internal Definitions . . . . . . . . . . . .   .   .   .   .   337
    3.3 Modeling with Mutable Data . . . . . . . . . . .    .   .   .   .   341
        3.3.1 Mutable List Structure . . . . . . . . . .    .   .   .   .   342
        3.3.2 Representing eues . . . . . . . . . . .      .   .   .   .   353
        3.3.3 Representing Tables . . . . . . . . . . .     .   .   .   .   360
        3.3.4 A Simulator for Digital Circuits . . . . .    .   .   .   .   369
        3.3.5 Propagation of Constraints . . . . . . .      .   .   .   .   386
    3.4 Concurrency: Time Is of the Essence . . . . . . .   .   .   .   .   401

          3.4.1 e Nature of Time in Concurrent Systems               .   .   403
          3.4.2 Mechanisms for Controlling Concurrency .              .   .   410
    3.5   Streams . . . . . . . . . . . . . . . . . . . . . . . . .   .   .   428
          3.5.1 Streams Are Delayed Lists . . . . . . . . . .         .   .   430
          3.5.2 Infinite Streams . . . . . . . . . . . . . . . .       .   .   441
          3.5.3 Exploiting the Stream Paradigm . . . . . . .          .   .   453
          3.5.4 Streams and Delayed Evaluation . . . . . .            .   .   470
          3.5.5 Modularity of Functional Programs
                 and Modularity of Objects . . . . . . . . . .        . .     479

4   Metalinguistic Abstraction                                                487
    4.1 e Metacircular Evaluator . . . . . . . . . . . . . . . .             492
         4.1.1 e Core of the Evaluator . . . . . . . . . . . .               495
         4.1.2 Representing Expressions . . . . . . . . . . . .               501
         4.1.3 Evaluator Data Structures . . . . . . . . . . . .              512
         4.1.4 Running the Evaluator as a Program . . . . . .                 518
         4.1.5 Data as Programs . . . . . . . . . . . . . . . . .             522
         4.1.6 Internal Definitions . . . . . . . . . . . . . . . .            526
         4.1.7 Separating Syntactic Analysis from Execution .                 534
    4.2 Variations on a Scheme — Lazy Evaluation . . . . . . .                541
         4.2.1 Normal Order and Applicative Order . . . . . .                 542
         4.2.2 An Interpreter with Lazy Evaluation . . . . . .                544
         4.2.3 Streams as Lazy Lists . . . . . . . . . . . . . . .            555
    4.3 Variations on a Scheme — Nondeterministic Computing                   559
         4.3.1 Amb and Search . . . . . . . . . . . . . . . . .               561
         4.3.2 Examples of Nondeterministic Programs . . . .                  567
         4.3.3 Implementing the amb Evaluator . . . . . . . .                 578
    4.4 Logic Programming . . . . . . . . . . . . . . . . . . . .             594
         4.4.1 Deductive Information Retrieval . . . . . . . .                599

         4.4.2    How the ery System Works . . . . . . .        .   .   615
         4.4.3    Is Logic Programming Mathematical Logic?       .   .   627
         4.4.4    Implementing the ery System . . . . . .       .   .   635
           e Driver Loop and Instantiation     .   .   636
           e Evaluator . . . . . . . . . . .   .   .   638
           Finding Assertions
                            by Paern Matching . . . . . . .     .   .   642
           Rules and Unification . . . . . . .   .   .   645
           Maintaining the Data Base . . . .    .   .   651
           Stream Operations . . . . . . . .    .   .   654
           ery Syntax Procedures . . . . .     .   .   656
           Frames and Bindings . . . . . . .    .   .   659

5   Computing with Register Maines                                      666
    5.1 Designing Register Machines . . . . . . . . . . . . . .      .   668
        5.1.1 A Language for Describing Register Machines            .   672
        5.1.2 Abstraction in Machine Design . . . . . . . .          .   678
        5.1.3 Subroutines . . . . . . . . . . . . . . . . . . .      .   681
        5.1.4 Using a Stack to Implement Recursion . . . .           .   686
        5.1.5 Instruction Summary . . . . . . . . . . . . . .        .   695
    5.2 A Register-Machine Simulator . . . . . . . . . . . . .       .   696
        5.2.1 e Machine Model . . . . . . . . . . . . . . .         .   698
        5.2.2 e Assembler . . . . . . . . . . . . . . . . .         .   704
        5.2.3 Generating Execution Procedures
                for Instructions . . . . . . . . . . . . . . . . .   .   708
        5.2.4 Monitoring Machine Performance . . . . . .             .   718
    5.3 Storage Allocation and Garbage Collection . . . . . .        .   723
        5.3.1 Memory as Vectors . . . . . . . . . . . . . . .        .   724
        5.3.2 Maintaining the Illusion of Infinite Memory .           .   731

   5.4   e Explicit-Control Evaluator . . . . . . . . . . . .     .   .   741
         5.4.1 e Core of the Explicit-Control Evaluator .         .   .   743
         5.4.2 Sequence Evaluation and Tail Recursion . .          .   .   751
         5.4.3 Conditionals, Assignments, and Definitions           .   .   756
         5.4.4 Running the Evaluator . . . . . . . . . . . .       .   .   759
   5.5   Compilation . . . . . . . . . . . . . . . . . . . . . .   .   .   767
         5.5.1 Structure of the Compiler . . . . . . . . . .       .   .   772
         5.5.2 Compiling Expressions . . . . . . . . . . . .       .   .   779
         5.5.3 Compiling Combinations . . . . . . . . . .          .   .   788
         5.5.4 Combining Instruction Sequences . . . . . .         .   .   797
         5.5.5 An Example of Compiled Code . . . . . . .           .   .   802
         5.5.6 Lexical Addressing . . . . . . . . . . . . . .      .   .   817
         5.5.7 Interfacing Compiled Code to the Evaluator          .   .   823

References                                                                 834

List of Exercises                                                          844

List of Figures                                                            846

Index                                                                      848

Colophon                                                                   855

Unofficial Texinfo Format

is is the second edition  book, from Unofficial Texinfo Format.
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But at least we don’t put our brave astronauts at risk by encoding the
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so think twice before you use your full name or distribute Info, ,
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Peath, Lytha Ayth

Addendum: See also the  video lectures by Abelson and Sussman:
at   or  .
Second Addendum: Above is the original introduction to the  from
2001. Ten years later,  has been transformed: mathematical symbols
and formulas are properly typeset, and figures drawn in vector graph-
ics. e original text formulas and  art figures are still there in

the Texinfo source, but will display only when compiled to Info output.
At the dawn of e-book readers and tablets, reading a  on screen is
officially not silly anymore. Enjoy!
A.R, May, 2011


T      , in respect and admiration, to the spirit that
    lives in the computer.

      “I think that it’s extraordinarily important that we in com-
      puter science keep fun in computing. When it started out,
      it was an awful lot of fun. Of course, the paying customers
      got shaed every now and then, and aer a while we began
      to take their complaints seriously. We began to feel as if we
      really were responsible for the successful, error-free perfect
      use of these machines. I don’t think we are. I think we’re
      responsible for stretching them, seing them off in new di-
      rections, and keeping fun in the house. I hope the field of
      computer science never loses its sense of fun. Above all, I
      hope we don’t become missionaries. Don’t feel as if you’re
      Bible salesmen. e world has too many of those already.
      What you know about computing other people will learn.
      Don’t feel as if the key to successful computing is only in
      your hands. What’s in your hands, I think and hope, is in-
      telligence: the ability to see the machine as more than when
      you were first led up to it, that you can make it more.”
      —Alan J. Perlis (April 1, 1922 – February 7, 1990)


E   , , , psychologists, and parents pro-
    gram. Armies, students, and some societies are programmed. An
assault on large problems employs a succession of programs, most of
which spring into existence en route. ese programs are rife with is-
sues that appear to be particular to the problem at hand. To appreciate
programming as an intellectual activity in its own right you must turn to
computer programming; you must read and write computer programs—
many of them. It doesn’t maer much what the programs are about or
what applications they serve. What does maer is how well they per-
form and how smoothly they fit with other programs in the creation
of still greater programs. e programmer must seek both perfection
of part and adequacy of collection. In this book the use of “program” is
focused on the creation, execution, and study of programs wrien in a
dialect of Lisp for execution on a digital computer. Using Lisp we re-
strict or limit not what we may program, but only the notation for our
program descriptions.
    Our traffic with the subject maer of this book involves us with
three foci of phenomena: the human mind, collections of computer pro-
grams, and the computer. Every computer program is a model, hatched
in the mind, of a real or mental process. ese processes, arising from

human experience and thought, are huge in number, intricate in de-
tail, and at any time only partially understood. ey are modeled to our
permanent satisfaction rarely by our computer programs. us even
though our programs are carefully handcraed discrete collections of
symbols, mosaics of interlocking functions, they continually evolve: we
change them as our perception of the model deepens, enlarges, gen-
eralizes until the model ultimately aains a metastable place within
still another model with which we struggle. e source of the exhilara-
tion associated with computer programming is the continual unfolding
within the mind and on the computer of mechanisms expressed as pro-
grams and the explosion of perception they generate. If art interprets
our dreams, the computer executes them in the guise of programs!
     For all its power, the computer is a harsh taskmaster. Its programs
must be correct, and what we wish to say must be said accurately in ev-
ery detail. As in every other symbolic activity, we become convinced of
program truth through argument. Lisp itself can be assigned a seman-
tics (another model, by the way), and if a program’s function can be
specified, say, in the predicate calculus, the proof methods of logic can
be used to make an acceptable correctness argument. Unfortunately, as
programs get large and complicated, as they almost always do, the ade-
quacy, consistency, and correctness of the specifications themselves be-
come open to doubt, so that complete formal arguments of correctness
seldom accompany large programs. Since large programs grow from
small ones, it is crucial that we develop an arsenal of standard program
structures of whose correctness we have become sure—we call them
idioms—and learn to combine them into larger structures using orga-
nizational techniques of proven value. ese techniques are treated at
length in this book, and understanding them is essential to participation
in the Promethean enterprise called programming. More than anything

else, the uncovering and mastery of powerful organizational techniques
accelerates our ability to create large, significant programs. Conversely,
since writing large programs is very taxing, we are stimulated to invent
new methods of reducing the mass of function and detail to be fied
into large programs.
     Unlike programs, computers must obey the laws of physics. If they
wish to perform rapidly—a few nanoseconds per state change—they
must transmit electrons only small distances (at most 1 12 feet). e heat
generated by the huge number of devices so concentrated in space has to
be removed. An exquisite engineering art has been developed balancing
between multiplicity of function and density of devices. In any event,
hardware always operates at a level more primitive than that at which
we care to program. e processes that transform our Lisp programs
to “machine” programs are themselves abstract models which we pro-
gram. eir study and creation give a great deal of insight into the or-
ganizational programs associated with programming arbitrary models.
Of course the computer itself can be so modeled. ink of it: the behav-
ior of the smallest physical switching element is modeled by quantum
mechanics described by differential equations whose detailed behavior
is captured by numerical approximations represented in computer pro-
grams executing on computers composed of . . .!
     It is not merely a maer of tactical convenience to separately iden-
tify the three foci. Even though, as they say, it’s all in the head, this
logical separation induces an acceleration of symbolic traffic between
these foci whose richness, vitality, and potential is exceeded in human
experience only by the evolution of life itself. At best, relationships be-
tween the foci are metastable. e computers are never large enough or
fast enough. Each breakthrough in hardware technology leads to more
massive programming enterprises, new organizational principles, and

an enrichment of abstract models. Every reader should ask himself pe-
riodically “Toward what end, toward what end?”—but do not ask it too
oen lest you pass up the fun of programming for the constipation of
biersweet philosophy.
     Among the programs we write, some (but never enough) perform a
precise mathematical function such as sorting or finding the maximum
of a sequence of numbers, determining primality, or finding the square
root. We call such programs algorithms, and a great deal is known of
their optimal behavior, particularly with respect to the two important
parameters of execution time and data storage requirements. A pro-
grammer should acquire good algorithms and idioms. Even though some
programs resist precise specifications, it is the responsibility of the pro-
grammer to estimate, and always to aempt to improve, their perfor-
     Lisp is a survivor, having been in use for about a quarter of a cen-
tury. Among the active programming languages only Fortran has had
a longer life. Both languages have supported the programming needs
of important areas of application, Fortran for scientific and engineering
computation and Lisp for artificial intelligence. ese two areas con-
tinue to be important, and their programmers are so devoted to these
two languages that Lisp and Fortran may well continue in active use for
at least another quarter-century.
     Lisp changes. e Scheme dialect used in this text has evolved from
the original Lisp and differs from the laer in several important ways,
including static scoping for variable binding and permiing functions to
yield functions as values. In its semantic structure Scheme is as closely
akin to Algol 60 as to early Lisps. Algol 60, never to be an active language
again, lives on in the genes of Scheme and Pascal. It would be difficult
to find two languages that are the communicating coin of two more dif-

ferent cultures than those gathered around these two languages. Pas-
cal is for building pyramids—imposing, breathtaking, static structures
built by armies pushing heavy blocks into place. Lisp is for building
organisms—imposing, breathtaking, dynamic structures built by squads
fiing fluctuating myriads of simpler organisms into place. e organiz-
ing principles used are the same in both cases, except for one extraordi-
narily important difference: e discretionary exportable functionality
entrusted to the individual Lisp programmer is more than an order of
magnitude greater than that to be found within Pascal enterprises. Lisp
programs inflate libraries with functions whose utility transcends the
application that produced them. e list, Lisp’s native data structure, is
largely responsible for such growth of utility. e simple structure and
natural applicability of lists are reflected in functions that are amazingly
nonidiosyncratic. In Pascal the plethora of declarable data structures in-
duces a specialization within functions that inhibits and penalizes ca-
sual cooperation. It is beer to have 100 functions operate on one data
structure than to have 10 functions operate on 10 data structures. As a
result the pyramid must stand unchanged for a millennium; the organ-
ism must evolve or perish.
     To illustrate this difference, compare the treatment of material and
exercises within this book with that in any first-course text using Pascal.
Do not labor under the illusion that this is a text digestible at  only,
peculiar to the breed found there. It is precisely what a serious book on
programming Lisp must be, no maer who the student is or where it is
     Note that this is a text about programming, unlike most Lisp books,
which are used as a preparation for work in artificial intelligence. Aer
all, the critical programming concerns of soware engineering and ar-
tificial intelligence tend to coalesce as the systems under investigation

become larger. is explains why there is such growing interest in Lisp
outside of artificial intelligence.
    As one would expect from its goals, artificial intelligence research
generates many significant programming problems. In other program-
ming cultures this spate of problems spawns new languages. Indeed, in
any very large programming task a useful organizing principle is to con-
trol and isolate traffic within the task modules via the invention of lan-
guage. ese languages tend to become less primitive as one approaches
the boundaries of the system where we humans interact most oen. As
a result, such systems contain complex language-processing functions
replicated many times. Lisp has such a simple syntax and semantics that
parsing can be treated as an elementary task. us parsing technology
plays almost no role in Lisp programs, and the construction of language
processors is rarely an impediment to the rate of growth and change of
large Lisp systems. Finally, it is this very simplicity of syntax and se-
mantics that is responsible for the burden and freedom borne by all
Lisp programmers. No Lisp program of any size beyond a few lines can
be wrien without being saturated with discretionary functions. Invent
and fit; have fits and reinvent! We toast the Lisp programmer who pens
his thoughts within nests of parentheses.
Alan J. Perlis
New Haven, Connecticut

Preface to the Second Edition

      Is it possible that soware is not like anything else, that it
      is meant to be discarded: that the whole point is to always
      see it as a soap bubble?
      —Alan J. Perlis

T        has been the basis of ’s entry-level
    computer science subject since 1980. We had been teaching this ma-
terial for four years when the first edition was published, and twelve
more years have elapsed until the appearance of this second edition.
We are pleased that our work has been widely adopted and incorpo-
rated into other texts. We have seen our students take the ideas and
programs in this book and build them in as the core of new computer
systems and languages. In literal realization of an ancient Talmudic pun,
our students have become our builders. We are lucky to have such ca-
pable students and such accomplished builders.
    In preparing this edition, we have incorporated hundreds of clarifi-
cations suggested by our own teaching experience and the comments of
colleagues at  and elsewhere. We have redesigned most of the ma-
jor programming systems in the book, including the generic-arithmetic
system, the interpreters, the register-machine simulator, and the com-
piler; and we have rewrien all the program examples to ensure that

any Scheme implementation conforming to the  Scheme standard
(IEEE 1990) will be able to run the code.
    is edition emphasizes several new themes. e most important of
these is the central role played by different approaches to dealing with
time in computational models: objects with state, concurrent program-
ming, functional programming, lazy evaluation, and nondeterministic
programming. We have included new sections on concurrency and non-
determinism, and we have tried to integrate this theme throughout the
    e first edition of the book closely followed the syllabus of our
 one-semester subject. With all the new material in the second edi-
tion, it will not be possible to cover everything in a single semester,
so the instructor will have to pick and choose. In our own teaching, we
sometimes skip the section on logic programming (Section 4.4), we have
students use the register-machine simulator but we do not cover its im-
plementation (Section 5.2), and we give only a cursory overview of the
compiler (Section 5.5). Even so, this is still an intense course. Some in-
structors may wish to cover only the first three or four chapters, leaving
the other material for subsequent courses.
    e World-Wide-Web site hp:// provides sup-
port for users of this book. is includes programs from the book, sam-
ple programming assignments, supplementary materials, and download-
able implementations of the Scheme dialect of Lisp.

Preface to the First Edition

      A computer is like a violin. You can imagine a novice try-
      ing first a phonograph and then a violin. e laer, he says,
      sounds terrible. at is the argument we have heard from
      our humanists and most of our computer scientists. Com-
      puter programs are good, they say, for particular purposes,
      but they aren’t flexible. Neither is a violin, or a typewriter,
      until you learn how to use it.
      —Marvin Minsky, “Why Programming Is a Good Medium
      for Expressing Poorly-Understood and Sloppily-Formulated

“T     S  I  C P”
      is the entry-level subject in computer science at the Massachuses
Institute of Technology. It is required of all students at  who major
in electrical engineering or in computer science, as one-fourth of the
“common core curriculum,” which also includes two subjects on circuits
and linear systems and a subject on the design of digital systems. We
have been involved in the development of this subject since 1978, and
we have taught this material in its present form since the fall of 1980 to
between 600 and 700 students each year. Most of these students have

had lile or no prior formal training in computation, although many
have played with computers a bit and a few have had extensive pro-
gramming or hardware-design experience.
    Our design of this introductory computer-science subject reflects
two major concerns. First, we want to establish the idea that a com-
puter language is not just a way of geing a computer to perform oper-
ations but rather that it is a novel formal medium for expressing ideas
about methodology. us, programs must be wrien for people to read,
and only incidentally for machines to execute. Second, we believe that
the essential material to be addressed by a subject at this level is not
the syntax of particular programming-language constructs, nor clever
algorithms for computing particular functions efficiently, nor even the
mathematical analysis of algorithms and the foundations of computing,
but rather the techniques used to control the intellectual complexity of
large soware systems.
    Our goal is that students who complete this subject should have a
good feel for the elements of style and the aesthetics of programming.
ey should have command of the major techniques for controlling
complexity in a large system. ey should be capable of reading a 50-
page-long program, if it is wrien in an exemplary style. ey should
know what not to read, and what they need not understand at any mo-
ment. ey should feel secure about modifying a program, retaining the
spirit and style of the original author.
    ese skills are by no means unique to computer programming. e
techniques we teach and draw upon are common to all of engineering
design. We control complexity by building abstractions that hide details
when appropriate. We control complexity by establishing conventional
interfaces that enable us to construct systems by combining standard,
well-understood pieces in a “mix and match” way. We control complex-

ity by establishing new languages for describing a design, each of which
emphasizes particular aspects of the design and deemphasizes others.
     Underlying our approach to this subject is our conviction that “com-
puter science” is not a science and that its significance has lile to do
with computers. e computer revolution is a revolution in the way we
think and in the way we express what we think. e essence of this
change is the emergence of what might best be called procedural epis-
temology —the study of the structure of knowledge from an imperative
point of view, as opposed to the more declarative point of view taken
by classical mathematical subjects. Mathematics provides a framework
for dealing precisely with notions of “what is.” Computation provides a
framework for dealing precisely with notions of “how to.”
     In teaching our material we use a dialect of the programming lan-
guage Lisp. We never formally teach the language, because we don’t
have to. We just use it, and students pick it up in a few days. is is
one great advantage of Lisp-like languages: ey have very few ways
of forming compound expressions, and almost no syntactic structure.
All of the formal properties can be covered in an hour, like the rules
of chess. Aer a short time we forget about syntactic details of the lan-
guage (because there are none) and get on with the real issues—figuring
out what we want to compute, how we will decompose problems into
manageable parts, and how we will work on the parts. Another advan-
tage of Lisp is that it supports (but does not enforce) more of the large-
scale strategies for modular decomposition of programs than any other
language we know. We can make procedural and data abstractions, we
can use higher-order functions to capture common paerns of usage,
we can model local state using assignment and data mutation, we can
link parts of a program with streams and delayed evaluation, and we can
easily implement embedded languages. All of this is embedded in an in-

teractive environment with excellent support for incremental program
design, construction, testing, and debugging. We thank all the genera-
tions of Lisp wizards, starting with John McCarthy, who have fashioned
a fine tool of unprecedented power and elegance.
    Scheme, the dialect of Lisp that we use, is an aempt to bring to-
gether the power and elegance of Lisp and Algol. From Lisp we take the
metalinguistic power that derives from the simple syntax, the uniform
representation of programs as data objects, and the garbage-collected
heap-allocated data. From Algol we take lexical scoping and block struc-
ture, which are gis from the pioneers of programming-language de-
sign who were on the Algol commiee. We wish to cite John Reynolds
and Peter Landin for their insights into the relationship of Church’s λ-
calculus to the structure of programming languages. We also recognize
our debt to the mathematicians who scouted out this territory decades
before computers appeared on the scene. ese pioneers include Alonzo
Church, Barkley Rosser, Stephen Kleene, and Haskell Curry.


W          the many people who have helped us
      develop this book and this curriculum.
    Our subject is a clear intellectual descendant of “6.231,” a wonderful
subject on programming linguistics and the λ-calculus taught at  in
the late 1960s by Jack Wozencra and Arthur Evans, Jr.
    We owe a great debt to Robert Fano, who reorganized ’s intro-
ductory curriculum in electrical engineering and computer science to
emphasize the principles of engineering design. He led us in starting
out on this enterprise and wrote the first set of subject notes from which
this book evolved.
    Much of the style and aesthetics of programming that we try to
teach were developed in conjunction with Guy Lewis Steele Jr., who
collaborated with Gerald Jay Sussman in the initial development of the
Scheme language. In addition, David Turner, Peter Henderson, Dan Fried-
man, David Wise, and Will Clinger have taught us many of the tech-
niques of the functional programming community that appear in this
    Joel Moses taught us about structuring large systems. His experi-
ence with the Macsyma system for symbolic computation provided the
insight that one should avoid complexities of control and concentrate

on organizing the data to reflect the real structure of the world being
    Marvin Minsky and Seymour Papert formed many of our aitudes
about programming and its place in our intellectual lives. To them we
owe the understanding that computation provides a means of expres-
sion for exploring ideas that would otherwise be too complex to deal
with precisely. ey emphasize that a student’s ability to write and
modify programs provides a powerful medium in which exploring be-
comes a natural activity.
    We also strongly agree with Alan Perlis that programming is lots of
fun and we had beer be careful to support the joy of programming. Part
of this joy derives from observing great masters at work. We are fortu-
nate to have been apprentice programmers at the feet of Bill Gosper and
Richard Greenbla.
    It is difficult to identify all the people who have contributed to the
development of our curriculum. We thank all the lecturers, recitation
instructors, and tutors who have worked with us over the past fieen
years and put in many extra hours on our subject, especially Bill Siebert,
Albert Meyer, Joe Stoy, Randy Davis, Louis Braida, Eric Grimson, Rod
Brooks, Lynn Stein and Peter Szolovits. We would like to specially ac-
knowledge the outstanding teaching contributions of Franklyn Turbak,
now at Wellesley; his work in undergraduate instruction set a standard
that we can all aspire to. We are grateful to Jerry Saltzer and Jim Miller
for helping us grapple with the mysteries of concurrency, and to Peter
Szolovits and David McAllester for their contributions to the exposition
of nondeterministic evaluation in Chapter 4.
    Many people have put in significant effort presenting this material
at other universities. Some of the people we have worked closely with
are Jacob Katzenelson at the Technion, Hardy Mayer at the University

of California at Irvine, Joe Stoy at Oxford, Elisha Sacks at Purdue, and
Jan Komorowski at the Norwegian University of Science and Technol-
ogy. We are exceptionally proud of our colleagues who have received
major teaching awards for their adaptations of this subject at other uni-
versities, including Kenneth Yip at Yale, Brian Harvey at the University
of California at Berkeley, and Dan Huenlocher at Cornell.
     Al Moyé arranged for us to teach this material to engineers at Hewle-
Packard, and for the production of videotapes of these lectures. We
would like to thank the talented instructors—in particular Jim Miller,
Bill Siebert, and Mike Eisenberg—who have designed continuing edu-
cation courses incorporating these tapes and taught them at universities
and industry all over the world.
     Many educators in other countries have put in significant work
translating the first edition. Michel Briand, Pierre Chamard, and An-
dré Pic produced a French edition; Susanne Daniels-Herold produced
a German edition; and Fumio Motoyoshi produced a Japanese edition.
We do not know who produced the Chinese edition, but we consider
it an honor to have been selected as the subject of an “unauthorized”
     It is hard to enumerate all the people who have made technical con-
tributions to the development of the Scheme systems we use for in-
structional purposes. In addition to Guy Steele, principal wizards have
included Chris Hanson, Joe Bowbeer, Jim Miller, Guillermo Rozas, and
Stephen Adams. Others who have put in significant time are Richard
Stallman, Alan Bawden, Kent Pitman, Jon Ta, Neil Mayle, John Lamp-
ing, Gwyn Osnos, Tracy Larrabee, George Carree, Soma Chaudhuri,
Bill Chiarchiaro, Steven Kirsch, Leigh Klotz, Wayne Noss, Todd Cass,
Patrick O’Donnell, Kevin eobald, Daniel Weise, Kenneth Sinclair, An-
thony Courtemanche, Henry M. Wu, Andrew Berlin, and Ruth Shyu.

    Beyond the  implementation, we would like to thank the many
people who worked on the  Scheme standard, including William
Clinger and Jonathan Rees, who edited the R4 RS, and Chris Haynes,
David Bartley, Chris Hanson, and Jim Miller, who prepared the 
    Dan Friedman has been a long-time leader of the Scheme commu-
nity. e community’s broader work goes beyond issues of language
design to encompass significant educational innovations, such as the
high-school curriculum based on EdScheme by Schemer’s Inc., and the
wonderful books by Mike Eisenberg and by Brian Harvey and Mahew
    We appreciate the work of those who contributed to making this a
real book, especially Terry Ehling, Larry Cohen, and Paul Bethge at the
 Press. Ella Mazel found the wonderful cover image. For the second
edition we are particularly grateful to Bernard and Ella Mazel for help
with the book design, and to David Jones, TEX wizard extraordinaire.
We also are indebted to those readers who made penetrating comments
on the new dra: Jacob Katzenelson, Hardy Mayer, Jim Miller, and es-
pecially Brian Harvey, who did unto this book as Julie did unto his book
Simply Scheme.
    Finally, we would like to acknowledge the support of the organiza-
tions that have encouraged this work over the years, including support
from Hewle-Packard, made possible by Ira Goldstein and Joel Birn-
baum, and support from , made possible by Bob Kahn.

Building Abstractions with Procedures

      e acts of the mind, wherein it exerts its power over simple
      ideas, are chiefly these three: 1. Combining several simple
      ideas into one compound one, and thus all complex ideas
      are made. 2. e second is bringing two ideas, whether sim-
      ple or complex, together, and seing them by one another
      so as to take a view of them at once, without uniting them
      into one, by which it gets all its ideas of relations. 3. e
      third is separating them from all other ideas that accom-
      pany them in their real existence: this is called abstraction,
      and thus all its general ideas are made.
      —John Locke, An Essay Concerning Human Understanding

W          the idea of a computational process. Com-
      putational processes are abstract beings that inhabit computers.
As they evolve, processes manipulate other abstract things called data.

e evolution of a process is directed by a paern of rules called a pro-
gram. People create programs to direct processes. In effect, we conjure
the spirits of the computer with our spells.
     A computational process is indeed much like a sorcerer’s idea of a
spirit. It cannot be seen or touched. It is not composed of maer at all.
However, it is very real. It can perform intellectual work. It can answer
questions. It can affect the world by disbursing money at a bank or by
controlling a robot arm in a factory. e programs we use to conjure
processes are like a sorcerer’s spells. ey are carefully composed from
symbolic expressions in arcane and esoteric programming languages that
prescribe the tasks we want our processes to perform.
     A computational process, in a correctly working computer, executes
programs precisely and accurately. us, like the sorcerer’s appren-
tice, novice programmers must learn to understand and to anticipate
the consequences of their conjuring. Even small errors (usually called
bugs or glitches) in programs can have complex and unanticipated con-
     Fortunately, learning to program is considerably less dangerous than
learning sorcery, because the spirits we deal with are conveniently con-
tained in a secure way. Real-world programming, however, requires
care, expertise, and wisdom. A small bug in a computer-aided design
program, for example, can lead to the catastrophic collapse of an air-
plane or a dam or the self-destruction of an industrial robot.
     Master soware engineers have the ability to organize programs so
that they can be reasonably sure that the resulting processes will per-
form the tasks intended. ey can visualize the behavior of their sys-
tems in advance. ey know how to structure programs so that unan-
ticipated problems do not lead to catastrophic consequences, and when
problems do arise, they can debug their programs. Well-designed com-

putational systems, like well-designed automobiles or nuclear reactors,
are designed in a modular manner, so that the parts can be constructed,
replaced, and debugged separately.

Programming in Lisp
We need an appropriate language for describing processes, and we will
use for this purpose the programming language Lisp. Just as our every-
day thoughts are usually expressed in our natural language (such as En-
glish, French, or Japanese), and descriptions of quantitative phenomena
are expressed with mathematical notations, our procedural thoughts
will be expressed in Lisp. Lisp was invented in the late 1950s as a for-
malism for reasoning about the use of certain kinds of logical expres-
sions, called recursion equations, as a model for computation. e lan-
guage was conceived by John McCarthy and is based on his paper “Re-
cursive Functions of Symbolic Expressions and eir Computation by
Machine” (McCarthy 1960).
     Despite its inception as a mathematical formalism, Lisp is a practi-
cal programming language. A Lisp interpreter is a machine that carries
out processes described in the Lisp language. e first Lisp interpreter
was implemented by McCarthy with the help of colleagues and stu-
dents in the Artificial Intelligence Group of the  Research Laboratory
of Electronics and in the  Computation Center.1 Lisp, whose name
is an acronym for LISt Processing, was designed to provide symbol-
manipulating capabilities for aacking programming problems such as
the symbolic differentiation and integration of algebraic expressions.
It included for this purpose new data objects known as atoms and lists,
     e Lisp 1 Programmer’s Manual appeared in 1960, and the Lisp 1.5 Programmer’s
Manual (McCarthy et al. 1965) was published in 1962. e early history of Lisp is de-
scribed in McCarthy 1978.

which most strikingly set it apart from all other languages of the period.
    Lisp was not the product of a concerted design effort. Instead, it
evolved informally in an experimental manner in response to users’
needs and to pragmatic implementation considerations. Lisp’s informal
evolution has continued through the years, and the community of Lisp
users has traditionally resisted aempts to promulgate any “official”
definition of the language. is evolution, together with the flexibility
and elegance of the initial conception, has enabled Lisp, which is the sec-
ond oldest language in widespread use today (only Fortran is older), to
continually adapt to encompass the most modern ideas about program
design. us, Lisp is by now a family of dialects, which, while sharing
most of the original features, may differ from one another in significant
ways. e dialect of Lisp used in this book is called Scheme.2
    Because of its experimental character and its emphasis on symbol
manipulation, Lisp was at first very inefficient for numerical compu-
tations, at least in comparison with Fortran. Over the years, however,
     e two dialects in which most major Lisp programs of the 1970s were wrien are
MacLisp (Moon 1978; Pitman 1983), developed at the  Project , and Interlisp
(Teitelman 1974), developed at Bolt Beranek and Newman Inc. and the Xerox Palo Alto
Research Center. Portable Standard Lisp (Hearn 1969; Griss 1981) was a Lisp dialect
designed to be easily portable between different machines. MacLisp spawned a number
of subdialects, such as Franz Lisp, which was developed at the University of California
at Berkeley, and Zetalisp (Moon and Weinreb 1981), which was based on a special-
purpose processor designed at the  Artificial Intelligence Laboratory to run Lisp
very efficiently. e Lisp dialect used in this book, called Scheme (Steele and Sussman
1975), was invented in 1975 by Guy Lewis Steele Jr. and Gerald Jay Sussman of the 
Artificial Intelligence Laboratory and later reimplemented for instructional use at .
Scheme became an  standard in 1990 (IEEE 1990). e Common Lisp dialect (Steele
1982, Steele 1990) was developed by the Lisp community to combine features from the
earlier Lisp dialects to make an industrial standard for Lisp. Common Lisp became an
 standard in 1994 (ANSI 1994).

Lisp compilers have been developed that translate programs into ma-
chine code that can perform numerical computations reasonably effi-
ciently. And for special applications, Lisp has been used with great ef-
fectiveness.3 Although Lisp has not yet overcome its old reputation as
hopelessly inefficient, Lisp is now used in many applications where ef-
ficiency is not the central concern. For example, Lisp has become a lan-
guage of choice for operating-system shell languages and for extension
languages for editors and computer-aided design systems.
    If Lisp is not a mainstream language, why are we using it as the
framework for our discussion of programming? Because the language
possesses unique features that make it an excellent medium for studying
important programming constructs and data structures and for relating
them to the linguistic features that support them. e most significant of
these features is the fact that Lisp descriptions of processes, called proce-
dures, can themselves be represented and manipulated as Lisp data. e
importance of this is that there are powerful program-design techniques
that rely on the ability to blur the traditional distinction between “pas-
sive” data and “active” processes. As we shall discover, Lisp’s flexibility
in handling procedures as data makes it one of the most convenient
languages in existence for exploring these techniques. e ability to
represent procedures as data also makes Lisp an excellent language for
writing programs that must manipulate other programs as data, such as
the interpreters and compilers that support computer languages. Above
and beyond these considerations, programming in Lisp is great fun.
     One such special application was a breakthrough computation of scientific
importance—an integration of the motion of the Solar System that extended previous
results by nearly two orders of magnitude, and demonstrated that the dynamics of the
Solar System is chaotic. is computation was made possible by new integration al-
gorithms, a special-purpose compiler, and a special-purpose computer all implemented
with the aid of soware tools wrien in Lisp (Abelson et al. 1992; Sussman and Wisdom

1.1 The Elements of Programming
A powerful programming language is more than just a means for in-
structing a computer to perform tasks. e language also serves as a
framework within which we organize our ideas about processes. us,
when we describe a language, we should pay particular aention to the
means that the language provides for combining simple ideas to form
more complex ideas. Every powerful language has three mechanisms
for accomplishing this:

       • primitive expressions, which represent the simplest entities the
         language is concerned with,

       • means of combination, by which compound elements are built
         from simpler ones, and

       • means of abstraction, by which compound elements can be named
         and manipulated as units.

In programming, we deal with two kinds of elements: procedures and
data. (Later we will discover that they are really not so distinct.) Infor-
mally, data is “stuff” that we want to manipulate, and procedures are
descriptions of the rules for manipulating the data. us, any powerful
programming language should be able to describe primitive data and
primitive procedures and should have methods for combining and ab-
stracting procedures and data.
    In this chapter we will deal only with simple numerical data so that
we can focus on the rules for building procedures.4 In later chapters we
     e characterization of numbers as “simple data” is a barefaced bluff. In fact, the
treatment of numbers is one of the trickiest and most confusing aspects of any pro-

will see that these same rules allow us to build procedures to manipulate
compound data as well.

1.1.1 Expressions
One easy way to get started at programming is to examine some typical
interactions with an interpreter for the Scheme dialect of Lisp. Imagine
that you are siing at a computer terminal. You type an expression, and
the interpreter responds by displaying the result of its evaluating that
    One kind of primitive expression you might type is a number. (More
precisely, the expression that you type consists of the numerals that
represent the number in base 10.) If you present Lisp with a number

the interpreter will respond by printing5

gramming language. Some typical issues involved are these: Some computer systems
distinguish integers, such as 2, from real numbers, such as 2.71. Is the real number 2.00
different from the integer 2? Are the arithmetic operations used for integers the same
as the operations used for real numbers? Does 6 divided by 2 produce 3, or 3.0? How
large a number can we represent? How many decimal places of accuracy can we repre-
sent? Is the range of integers the same as the range of real numbers? Above and beyond
these questions, of course, lies a collection of issues concerning roundoff and trunca-
tion errors—the entire science of numerical analysis. Since our focus in this book is on
large-scale program design rather than on numerical techniques, we are going to ignore
these problems. e numerical examples in this chapter will exhibit the usual roundoff
behavior that one observes when using arithmetic operations that preserve a limited
number of decimal places of accuracy in noninteger operations.
      roughout this book, when we wish to emphasize the distinction between the
input typed by the user and the response printed by the interpreter, we will show the
laer in slanted characters.

Expressions representing numbers may be combined with an expres-
sion representing a primitive procedure (such as + or *) to form a com-
pound expression that represents the application of the procedure to
those numbers. For example:
(+ 137 349)

(- 1000 334)

(* 5 99)

(/ 10 5)

(+ 2.7 10)

Expressions such as these, formed by delimiting a list of expressions
within parentheses in order to denote procedure application, are called
combinations. e lemost element in the list is called the operator, and
the other elements are called operands. e value of a combination is
obtained by applying the procedure specified by the operator to the ar-
guments that are the values of the operands.
     e convention of placing the operator to the le of the operands
is known as prefix notation, and it may be somewhat confusing at first
because it departs significantly from the customary mathematical con-
vention. Prefix notation has several advantages, however. One of them
is that it can accommodate procedures that may take an arbitrary num-
ber of arguments, as in the following examples:

(+ 21 35 12 7)

(* 25 4 12)

No ambiguity can arise, because the operator is always the lemost el-
ement and the entire combination is delimited by the parentheses.
    A second advantage of prefix notation is that it extends in a straight-
forward way to allow combinations to be nested, that is, to have combi-
nations whose elements are themselves combinations:
(+ (* 3 5) (- 10 6))

ere is no limit (in principle) to the depth of such nesting and to the
overall complexity of the expressions that the Lisp interpreter can eval-
uate. It is we humans who get confused by still relatively simple expres-
sions such as
(+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))

which the interpreter would readily evaluate to be 57. We can help our-
selves by writing such an expression in the form
(+ (* 3
        (+ (* 2 4)
           (+ 3 5)))
     (+ (- 10 7)

following a formaing convention known as prey-printing, in which
each long combination is wrien so that the operands are aligned ver-
tically. e resulting indentations display clearly the structure of the

    Even with complex expressions, the interpreter always operates in
the same basic cycle: It reads an expression from the terminal, evaluates
the expression, and prints the result. is mode of operation is oen
expressed by saying that the interpreter runs in a read-eval-print loop.
Observe in particular that it is not necessary to explicitly instruct the
interpreter to print the value of the expression.7

1.1.2 Naming and the Environment
A critical aspect of a programming language is the means it provides
for using names to refer to computational objects. We say that the name
identifies a variable whose value is the object.
    In the Scheme dialect of Lisp, we name things with define. Typing
(define size 2)

causes the interpreter to associate the value 2 with the name size.8
Once the name size has been associated with the number 2, we can
refer to the value 2 by name:

      Lisp systems typically provide features to aid the user in formaing expressions.
Two especially useful features are one that automatically indents to the proper prey-
print position whenever a new line is started and one that highlights the matching le
parenthesis whenever a right parenthesis is typed.
      Lisp obeys the convention that every expression has a value. is convention, to-
gether with the old reputation of Lisp as an inefficient language, is the source of the
quip by Alan Perlis (paraphrasing Oscar Wilde) that “Lisp programmers know the value
of everything but the cost of nothing.”
      In this book, we do not show the interpreter’s response to evaluating definitions,
since this is highly implementation-dependent.

(* 5 size)

Here are further examples of the use of define:
(define pi 3.14159)
(define radius 10)
(* pi (* radius radius))
(define circumference (* 2 pi radius))

define    is our language’s simplest means of abstraction, for it allows
us to use simple names to refer to the results of compound operations,
such as the circumference computed above. In general, computational
objects may have very complex structures, and it would be extremely
inconvenient to have to remember and repeat their details each time we
want to use them. Indeed, complex programs are constructed by build-
ing, step by step, computational objects of increasing complexity. e
interpreter makes this step-by-step program construction particularly
convenient because name-object associations can be created incremen-
tally in successive interactions. is feature encourages the incremental
development and testing of programs and is largely responsible for the
fact that a Lisp program usually consists of a large number of relatively
simple procedures.
    It should be clear that the possibility of associating values with sym-
bols and later retrieving them means that the interpreter must maintain
some sort of memory that keeps track of the name-object pairs. is
memory is called the environment (more precisely the global environ-
ment, since we will see later that a computation may involve a number

of different environments).9

1.1.3 Evaluating Combinations
One of our goals in this chapter is to isolate issues about thinking pro-
cedurally. As a case in point, let us consider that, in evaluating combi-
nations, the interpreter is itself following a procedure.
    To evaluate a combination, do the following:

   1. Evaluate the subexpressions of the combination.

   2. Apply the procedure that is the value of the lemost subexpres-
      sion (the operator) to the arguments that are the values of the
      other subexpressions (the operands).

Even this simple rule illustrates some important points about processes
in general. First, observe that the first step dictates that in order to ac-
complish the evaluation process for a combination we must first per-
form the evaluation process on each element of the combination. us,
the evaluation rule is recursive in nature; that is, it includes, as one of
its steps, the need to invoke the rule itself.10
     Notice how succinctly the idea of recursion can be used to express
what, in the case of a deeply nested combination, would otherwise be
viewed as a rather complicated process. For example, evaluating
      Chapter 3 will show that this notion of environment is crucial, both for under-
standing how the interpreter works and for implementing interpreters.
      It may seem strange that the evaluation rule says, as part of the first step, that
we should evaluate the lemost element of a combination, since at this point that can
only be an operator such as + or * representing a built-in primitive procedure such as
addition or multiplication. We will see later that it is useful to be able to work with
combinations whose operators are themselves compound expressions.


                          * 26            15

                           + 2 24
                                         + 3 5 7

                               * 4 6

      Figure 1.1: Tree representation, showing the value of each

(* (+ 2 (* 4 6))
   (+ 3 5 7))

requires that the evaluation rule be applied to four different combina-
tions. We can obtain a picture of this process by representing the combi-
nation in the form of a tree, as shown in Figure 1.1. Each combination is
represented by a node with branches corresponding to the operator and
the operands of the combination stemming from it. e terminal nodes
(that is, nodes with no branches stemming from them) represent either
operators or numbers. Viewing evaluation in terms of the tree, we can
imagine that the values of the operands percolate upward, starting from
the terminal nodes and then combining at higher and higher levels. In
general, we shall see that recursion is a very powerful technique for
dealing with hierarchical, treelike objects. In fact, the “percolate values
upward” form of the evaluation rule is an example of a general kind of
process known as tree accumulation.
    Next, observe that the repeated application of the first step brings us
to the point where we need to evaluate, not combinations, but primitive
expressions such as numerals, built-in operators, or other names. We

take care of the primitive cases by stipulating that

    • the values of numerals are the numbers that they name,

    • the values of built-in operators are the machine instruction se-
      quences that carry out the corresponding operations, and

    • the values of other names are the objects associated with those
      names in the environment.

We may regard the second rule as a special case of the third one by stip-
ulating that symbols such as + and * are also included in the global envi-
ronment, and are associated with the sequences of machine instructions
that are their “values.” e key point to notice is the role of the environ-
ment in determining the meaning of the symbols in expressions. In an
interactive language such as Lisp, it is meaningless to speak of the value
of an expression such as (+ x 1) without specifying any information
about the environment that would provide a meaning for the symbol
x (or even for the symbol +). As we shall see in Chapter 3, the general
notion of the environment as providing a context in which evaluation
takes place will play an important role in our understanding of program
    Notice that the evaluation rule given above does not handle defini-
tions. For instance, evaluating (define x 3) does not apply define to
two arguments, one of which is the value of the symbol x and the other
of which is 3, since the purpose of the define is precisely to associate x
with a value. (at is, (define x 3) is not a combination.)
    Such exceptions to the general evaluation rule are called special forms.
define is the only example of a special form that we have seen so far,
but we will meet others shortly. Each special form has its own evalu-
ation rule. e various kinds of expressions (each with its associated

evaluation rule) constitute the syntax of the programming language. In
comparison with most other programming languages, Lisp has a very
simple syntax; that is, the evaluation rule for expressions can be de-
scribed by a simple general rule together with specialized rules for a
small number of special forms.11

1.1.4 Compound Procedures
We have identified in Lisp some of the elements that must appear in any
powerful programming language:

       • Numbers and arithmetic operations are primitive data and proce-

       • Nesting of combinations provides a means of combining opera-

       • Definitions that associate names with values provide a limited
         means of abstraction.

Now we will learn about procedure definitions, a much more powerful
abstraction technique by which a compound operation can be given a
name and then referred to as a unit.
     Special syntactic forms that are simply convenient alternative surface structures
for things that can be wrien in more uniform ways are sometimes called syntactic
sugar, to use a phrase coined by Peter Landin. In comparison with users of other lan-
guages, Lisp programmers, as a rule, are less concerned with maers of syntax. (By
contrast, examine any Pascal manual and notice how much of it is devoted to descrip-
tions of syntax.) is disdain for syntax is due partly to the flexibility of Lisp, which
makes it easy to change surface syntax, and partly to the observation that many “con-
venient” syntactic constructs, which make the language less uniform, end up causing
more trouble than they are worth when programs become large and complex. In the
words of Alan Perlis, “Syntactic sugar causes cancer of the semicolon.”

    We begin by examining how to express the idea of “squaring.” We
might say, “To square something, multiply it by itself.” is is expressed
in our language as
(define (square x) (* x x))

We can understand this in the following way:
(define (square           x)             (*         x             x))
  |            |          |                |        |             |
 To        square     something,      multiply      it    by   itself.

We have here a compound procedure, which has been given the name
square. e procedure represents the operation of multiplying some-
thing by itself. e thing to be multiplied is given a local name, x, which
plays the same role that a pronoun plays in natural language. Evaluating
the definition creates this compound procedure and associates it with
the name square.12
    e general form of a procedure definition is
(define (⟨name⟩      ⟨formal parameters⟩)
  ⟨ body⟩)
e ⟨name ⟩ is a symbol to be associated with the procedure definition in
the environment.13 e ⟨formal parameters ⟩ are the names used within
the body of the procedure to refer to the corresponding arguments of
the procedure. e ⟨body ⟩ is an expression that will yield the value of
      Observe that there are two different operations being combined here: we are creat-
ing the procedure, and we are giving it the name square. It is possible, indeed important,
to be able to separate these two notions—to create procedures without naming them,
and to give names to procedures that have already been created. We will see how to do
this in Section 1.3.2.
   13 roughout this book, we will describe the general syntax of expressions by using

italic symbols delimited by angle brackets—e.g., ⟨name ⟩—to denote the “slots” in the
expression to be filled in when such an expression is actually used.

the procedure application when the formal parameters are replaced by
the actual arguments to which the procedure is applied.14 e ⟨name ⟩
and the ⟨formal parameters ⟩ are grouped within parentheses, just as they
would be in an actual call to the procedure being defined.
    Having defined square, we can now use it:
(square 21)
(square (+ 2 5))
(square (square 3))

We can also use square as a building block in defining other procedures.
For example, x 2 + y 2 can be expressed as
(+ (square x) (square y))

We can easily define a procedure sum-of-squares that, given any two
numbers as arguments, produces the sum of their squares:
(define (sum-of-squares x y)
  (+ (square x) (square y)))
(sum-of-squares 3 4)

Now we can use sum-of-squares as a building block in constructing
further procedures:
(define (f a)
  (sum-of-squares (+ a 1) (* a 2)))
(f 5)

  14 More generally, the body of the procedure can be a sequence of expressions. In this

case, the interpreter evaluates each expression in the sequence in turn and returns the
value of the final expression as the value of the procedure application.

Compound procedures are used in exactly the same way as primitive
procedures. Indeed, one could not tell by looking at the definition of
sum-of-squares given above whether square was built into the inter-
preter, like + and *, or defined as a compound procedure.

1.1.5 The Substitution Model for Procedure Application
To evaluate a combination whose operator names a compound proce-
dure, the interpreter follows much the same process as for combina-
tions whose operators name primitive procedures, which we described
in Section 1.1.3. at is, the interpreter evaluates the elements of the
combination and applies the procedure (which is the value of the oper-
ator of the combination) to the arguments (which are the values of the
operands of the combination).
    We can assume that the mechanism for applying primitive proce-
dures to arguments is built into the interpreter. For compound proce-
dures, the application process is as follows:

        To apply a compound procedure to arguments, evaluate the
        body of the procedure with each formal parameter replaced
        by the corresponding argument.

To illustrate this process, let’s evaluate the combination
(f 5)

where f is the procedure defined in Section 1.1.4. We begin by retrieving
the body of f:
(sum-of-squares (+ a 1) (* a 2))

en we replace the formal parameter a by the argument 5:
(sum-of-squares (+ 5 1) (* 5 2))

us the problem reduces to the evaluation of a combination with two
operands and an operator sum-of-squares. Evaluating this combina-
tion involves three subproblems. We must evaluate the operator to get
the procedure to be applied, and we must evaluate the operands to get
the arguments. Now (+ 5 1) produces 6 and (* 5 2) produces 10, so
we must apply the sum-of-squares procedure to 6 and 10. ese values
are substituted for the formal parameters x and y in the body of sum-
of-squares, reducing the expression to
(+ (square 6) (square 10))

If we use the definition of square, this reduces to
(+ (* 6 6) (* 10 10))

which reduces by multiplication to
(+ 36 100)

and finally to

e process we have just described is called the substitution model for
procedure application. It can be taken as a model that determines the
“meaning” of procedure application, insofar as the procedures in this
chapter are concerned. However, there are two points that should be

      • e purpose of the substitution is to help us think about proce-
        dure application, not to provide a description of how the inter-
        preter really works. Typical interpreters do not evaluate proce-
        dure applications by manipulating the text of a procedure to sub-
        stitute values for the formal parameters. In practice, the “substi-
        tution” is accomplished by using a local environment for the for-
        mal parameters. We will discuss this more fully in Chapter 3 and

       Chapter 4 when we examine the implementation of an interpreter
       in detail.

    • Over the course of this book, we will present a sequence of in-
      creasingly elaborate models of how interpreters work, culminat-
      ing with a complete implementation of an interpreter and com-
      piler in Chapter 5. e substitution model is only the first of these
      models—a way to get started thinking formally about the evalu-
      ation process. In general, when modeling phenomena in science
      and engineering, we begin with simplified, incomplete models.
      As we examine things in greater detail, these simple models be-
      come inadequate and must be replaced by more refined models.
      e substitution model is no exception. In particular, when we
      address in Chapter 3 the use of procedures with “mutable data,”
      we will see that the substitution model breaks down and must be
      replaced by a more complicated model of procedure application.15

Applicative order versus normal order
According to the description of evaluation given in Section 1.1.3, the
interpreter first evaluates the operator and operands and then applies
the resulting procedure to the resulting arguments. is is not the only
way to perform evaluation. An alternative evaluation model would not
evaluate the operands until their values were needed. Instead it would
  15 Despite the simplicity of the substitution idea, it turns out to be surprisingly com-

plicated to give a rigorous mathematical definition of the substitution process. e
problem arises from the possibility of confusion between the names used for the formal
parameters of a procedure and the (possibly identical) names used in the expressions to
which the procedure may be applied. Indeed, there is a long history of erroneous def-
initions of substitution in the literature of logic and programming semantics. See Stoy
1977 for a careful discussion of substitution.

first substitute operand expressions for parameters until it obtained an
expression involving only primitive operators, and would then perform
the evaluation. If we used this method, the evaluation of (f 5) would
proceed according to the sequence of expansions
(sum-of-squares (+ 5 1) (* 5 2))
(+   (square (+ 5 1))          (square (* 5 2))   )
(+   (* (+ 5 1) (+ 5 1))       (* (* 5 2) (* 5 2)))

followed by the reductions
(+       (* 6 6)         (* 10 10))
(+          36              100)

is gives the same answer as our previous evaluation model, but the
process is different. In particular, the evaluations of (+ 5 1) and (* 5
2) are each performed twice here, corresponding to the reduction of the
expression (* x x) with x replaced respectively by (+ 5 1) and (* 5
    is alternative “fully expand and then reduce” evaluation method
is known as normal-order evaluation, in contrast to the “evaluate the
arguments and then apply” method that the interpreter actually uses,
which is called applicative-order evaluation. It can be shown that, for
procedure applications that can be modeled using substitution (includ-
ing all the procedures in the first two chapters of this book) and that
yield legitimate values, normal-order and applicative-order evaluation
produce the same value. (See Exercise 1.5 for an instance of an “illegit-
imate” value where normal-order and applicative-order evaluation do
not give the same result.)
    Lisp uses applicative-order evaluation, partly because of the addi-
tional efficiency obtained from avoiding multiple evaluations of expres-
sions such as those illustrated with (+ 5 1) and (* 5 2) above and, more

significantly, because normal-order evaluation becomes much more com-
plicated to deal with when we leave the realm of procedures that can be
modeled by substitution. On the other hand, normal-order evaluation
can be an extremely valuable tool, and we will investigate some of its
implications in Chapter 3 and Chapter 4.16

1.1.6 Conditional Expressions and Predicates
e expressive power of the class of procedures that we can define at
this point is very limited, because we have no way to make tests and
to perform different operations depending on the result of a test. For
instance, we cannot define a procedure that computes the absolute value
of a number by testing whether the number is positive, negative, or zero
and taking different actions in the different cases according to the rule
                                 x if x > 0,
                        |x | = 
                                 0 if x = 0,
                                −x if x < 0.

is construct is called a case analysis, and there is a special form in
Lisp for notating such a case analysis. It is called cond (which stands for
“conditional”), and it is used as follows:
(define (abs x)
  (cond ((> x 0) x)
          ((= x 0) 0)
          ((< x 0) (- x))))

e general form of a conditional expression is
  16 In Chapter 3 we will introduce stream processing, which is a way of handling appar-

ently “infinite” data structures by incorporating a limited form of normal-order evalu-
ation. In Section 4.2 we will modify the Scheme interpreter to produce a normal-order
variant of Scheme.

(cond (⟨p1 ⟩     ⟨e1 ⟩)
        (⟨p2 ⟩   ⟨e2 ⟩)
        (⟨pn ⟩ ⟨en ⟩))

consisting of the symbol cond followed by parenthesized pairs of ex-
(⟨p⟩   ⟨e⟩)
called clauses. e first expression in each pair is a predicate—that is, an
expression whose value is interpreted as either true or false.17
     Conditional expressions are evaluated as follows. e predicate ⟨p 1 ⟩
is evaluated first. If its value is false, then ⟨p 2 ⟩ is evaluated. If ⟨p 2 ⟩’s
value is also false, then ⟨p 3 ⟩ is evaluated. is process continues until
a predicate is found whose value is true, in which case the interpreter
returns the value of the corresponding consequent expression ⟨e⟩ of the
clause as the value of the conditional expression. If none of the ⟨p⟩’s is
found to be true, the value of the cond is undefined.
     e word predicate is used for procedures that return true or false,
as well as for expressions that evaluate to true or false. e absolute-
value procedure abs makes use of the primitive predicates >, <, and =.18
ese take two numbers as arguments and test whether the first number
is, respectively, greater than, less than, or equal to the second number,
returning true or false accordingly.
     Another way to write the absolute-value procedure is
  17 “Interpreted  as either true or false” means this: In Scheme, there are two distin-
guished values that are denoted by the constants #t and #f. When the interpreter checks
a predicate’s value, it interprets #f as false. Any other value is treated as true. (us,
providing #t is logically unnecessary, but it is convenient.) In this book we will use
names true and false, which are associated with the values #t and #f respectively.
   18 abs also uses the “minus” operator -, which, when used with a single operand, as

in (- x), indicates negation.

(define (abs x)
  (cond ((< x 0) (- x))
           (else x)))

which could be expressed in English as “If x is less than zero return −x;
otherwise return x.” else is a special symbol that can be used in place of
the ⟨p⟩ in the final clause of a cond. is causes the cond to return as its
value the value of the corresponding ⟨e⟩ whenever all previous clauses
have been bypassed. In fact, any expression that always evaluates to a
true value could be used as the ⟨p⟩ here.
    Here is yet another way to write the absolute-value procedure:
(define (abs x)
  (if (< x 0)
        (- x)

is uses the special form if, a restricted type of conditional that can
be used when there are precisely two cases in the case analysis. e
general form of an if expression is
(if   ⟨ predicate ⟩ ⟨ consequent ⟩ ⟨ alternative ⟩)
To evaluate an if expression, the interpreter starts by evaluating the
⟨predicate ⟩ part of the expression. If the ⟨predicate ⟩ evaluates to a true
value, the interpreter then evaluates the ⟨consequent ⟩ and returns its
value. Otherwise it evaluates the ⟨alternative ⟩ and returns its value.19
    In addition to primitive predicates such as <, =, and >, there are log-
ical composition operations, which enable us to construct compound
  19 A minor difference between if and cond is that the ⟨e⟩ part of each cond clause may

be a sequence of expressions. If the corresponding ⟨p⟩ is found to be true, the expres-
sions ⟨e⟩ are evaluated in sequence and the value of the final expression in the sequence
is returned as the value of the cond. In an if expression, however, the ⟨consequent ⟩ and
⟨alternative ⟩ must be single expressions.

predicates. e three most frequently used are these:

    • (and ⟨e 1 ⟩ . . . ⟨en ⟩)
      e interpreter evaluates the expressions ⟨e ⟩ one at a time, in le-
      to-right order. If any ⟨e ⟩ evaluates to false, the value of the and
      expression is false, and the rest of the ⟨e ⟩’s are not evaluated. If
      all ⟨e ⟩’s evaluate to true values, the value of the and expression is
      the value of the last one.

    • (or ⟨e 1 ⟩ . . . ⟨en ⟩)
      e interpreter evaluates the expressions ⟨e ⟩ one at a time, in le-
      to-right order. If any ⟨e ⟩ evaluates to a true value, that value is
      returned as the value of the or expression, and the rest of the
      ⟨e ⟩’s are not evaluated. If all ⟨e ⟩’s evaluate to false, the value of
      the or expression is false.

    • (not ⟨e⟩)
      e value of a not expression is true when the expression ⟨e ⟩
      evaluates to false, and false otherwise.

Notice that and and or are special forms, not procedures, because the
subexpressions are not necessarily all evaluated. not is an ordinary pro-
    As an example of how these are used, the condition that a number
x be in the range 5 < x < 10 may be expressed as
(and (> x 5) (< x 10))

As another example, we can define a predicate to test whether one num-
ber is greater than or equal to another as
(define (>= x y) (or (> x y) (= x y)))

or alternatively as
(define (>= x y) (not (< x y)))

      Exercise 1.1: Below is a sequence of expressions. What is
      the result printed by the interpreter in response to each ex-
      pression? Assume that the sequence is to be evaluated in
      the order in which it is presented.
      (+ 5 3 4)
      (- 9 1)
      (/ 6 2)
      (+ (* 2 4) (- 4 6))
      (define a 3)
      (define b (+ a 1))
      (+ a b (* a b))
      (= a b)
      (if (and (> b a) (< b (* a b)))

      (cond ((= a 4) 6)
               ((= b 4) (+ 6 7 a))
               (else 25))

      (+ 2 (if (> b a) b a))

      (* (cond ((> a b) a)
                 ((< a b) b)
                 (else -1))
           (+ a 1))

Exercise 1.2: Translate the following expression into prefix
                 5 + 4 + (2 − (3 − (6 + 54 )))
                       3(6 − 2)(2 − 7)
Exercise 1.3: Define a procedure that takes three numbers
as arguments and returns the sum of the squares of the two
larger numbers.

Exercise 1.4: Observe that our model of evaluation allows
for combinations whose operators are compound expres-
sions. Use this observation to describe the behavior of the
following procedure:
(define (a-plus-abs-b a b)
  ((if (> b 0) + -) a b))

Exercise 1.5: Ben Bitdiddle has invented a test to determine
whether the interpreter he is faced with is using applicative-
order evaluation or normal-order evaluation. He defines the
following two procedures:
(define (p) (p))
(define (test x y)
  (if (= x 0) 0 y))

en he evaluates the expression
(test 0 (p))

What behavior will Ben observe with an interpreter that
uses applicative-order evaluation? What behavior will he
observe with an interpreter that uses normal-order evalu-
ation? Explain your answer. (Assume that the evaluation

      rule for the special form if is the same whether the in-
      terpreter is using normal or applicative order: e predi-
      cate expression is evaluated first, and the result determines
      whether to evaluate the consequent or the alternative ex-

1.1.7 Example: Square Roots by Newton’s Method
Procedures, as introduced above, are much like ordinary mathematical
functions. ey specify a value that is determined by one or more pa-
rameters. But there is an important difference between mathematical
functions and computer procedures. Procedures must be effective.
   As a case in point, consider the problem of computing square roots.
We can define the square-root function as
             x = the y such that y ≥ 0 and y 2 = x .

is describes a perfectly legitimate mathematical function. We could
use it to recognize whether one number is the square root of another,
or to derive facts about square roots in general. On the other hand, the
definition does not describe a procedure. Indeed, it tells us almost noth-
ing about how to actually find the square root of a given number. It will
not help maers to rephrase this definition in pseudo-Lisp:
(define (sqrt x)
  (the y (and (>= y 0)
               (= (square y) x))))

is only begs the question.
    e contrast between function and procedure is a reflection of the
general distinction between describing properties of things and describ-
ing how to do things, or, as it is sometimes referred to, the distinction

between declarative knowledge and imperative knowledge. In mathe-
matics we are usually concerned with declarative (what is) descriptions,
whereas in computer science we are usually concerned with imperative
(how to) descriptions.20
     How does one compute square roots? e most common way is to
use Newton’s method of successive approximations, which says that
whenever we have a guess y for the value of the square root of a number
x , we can perform a simple manipulation to get a beer guess (one closer
to the actual square root) by averaging y with x/y.21 For example, we
can compute the square root of 2 as follows. Suppose our initial guess
is 1:
Guess         Quotient                       Average
1             (2/1) = 2                      ((2 + 1)/2) = 1.5
1.5           (2/1.5) = 1.3333               ((1.3333 + 1.5)/2) = 1.4167
1.4167        (2/1.4167) = 1.4118            ((1.4167 + 1.4118)/2) = 1.4142
1.4142        ...                            ...
   20 Declarative and imperative descriptions are intimately related, as indeed are math-

ematics and computer science. For instance, to say that the answer produced by a pro-
gram is “correct” is to make a declarative statement about the program. ere is a large
amount of research aimed at establishing techniques for proving that programs are
correct, and much of the technical difficulty of this subject has to do with negotiating
the transition between imperative statements (from which programs are constructed)
and declarative statements (which can be used to deduce things). In a related vein, an
important current area in programming-language design is the exploration of so-called
very high-level languages, in which one actually programs in terms of declarative state-
ments. e idea is to make interpreters sophisticated enough so that, given “what is”
knowledge specified by the programmer, they can generate “how to” knowledge auto-
matically. is cannot be done in general, but there are important areas where progress
has been made. We shall revisit this idea in Chapter 4.
   21 is square-root algorithm is actually a special case of Newton’s method, which is

a general technique for finding roots of equations. e square-root algorithm itself was
developed by Heron of Alexandria in the first century .. We will see how to express
the general Newton’s method as a Lisp procedure in Section 1.3.4.

Continuing this process, we obtain beer and beer approximations to
the square root.
    Now let’s formalize the process in terms of procedures. We start
with a value for the radicand (the number whose square root we are
trying to compute) and a value for the guess. If the guess is good enough
for our purposes, we are done; if not, we must repeat the process with
an improved guess. We write this basic strategy as a procedure:
(define (sqrt-iter guess x)
  (if (good-enough? guess x)
        (sqrt-iter (improve guess x) x)))

A guess is improved by averaging it with the quotient of the radicand
and the old guess:
(define (improve guess x)
  (average guess (/ x guess)))

(define (average x y)
  (/ (+ x y) 2))

We also have to say what we mean by “good enough.” e following
will do for illustration, but it is not really a very good test. (See Exercise
1.7.) e idea is to improve the answer until it is close enough so that its
square differs from the radicand by less than a predetermined tolerance
(here 0.001):22
(define (good-enough? guess x)
  (< (abs (- (square guess) x)) 0.001))

  22 We will usually give predicates names ending with question marks, to help us re-

member that they are predicates. is is just a stylistic convention. As far as the inter-
preter is concerned, the question mark is just an ordinary character.

Finally, we need a way to get started. For instance, we can always guess
that the square root of any number is 1:23
(define (sqrt x)
  (sqrt-iter 1.0 x))

If we type these definitions to the interpreter, we can use sqrt just as
we can use any procedure:
(sqrt 9)

(sqrt (+ 100 37))

(sqrt (+ (sqrt 2) (sqrt 3)))

(square (sqrt 1000))

e sqrt program also illustrates that the simple procedural language
we have introduced so far is sufficient for writing any purely numeri-
cal program that one could write in, say, C or Pascal. is might seem
surprising, since we have not included in our language any iterative
   23 Observe that we express our initial guess as 1.0 rather than 1. is would not make

any difference in many Lisp implementations.  Scheme, however, distinguishes be-
tween exact integers and decimal values, and dividing two integers produces a rational
number rather than a decimal. For example, dividing 10 by 6 yields 5/3, while dividing
10.0 by 6.0 yields 1.6666666666666667. (We will learn how to implement arithmetic on
rational numbers in Section 2.1.1.) If we start with an initial guess of 1 in our square-root
program, and x is an exact integer, all subsequent values produced in the square-root
computation will be rational numbers rather than decimals. Mixed operations on ratio-
nal numbers and decimals always yield decimals, so starting with an initial guess of 1.0
forces all subsequent values to be decimals.

(looping) constructs that direct the computer to do something over and
over again. sqrt-iter, on the other hand, demonstrates how iteration
can be accomplished using no special construct other than the ordinary
ability to call a procedure.24

       Exercise 1.6: Alyssa P. Hacker doesn’t see why if needs to
       be provided as a special form. “Why can’t I just define it as
       an ordinary procedure in terms of cond?” she asks. Alyssa’s
       friend Eva Lu Ator claims this can indeed be done, and she
       defines a new version of if:
       (define (new-if predicate then-clause else-clause)
           (cond (predicate then-clause)
                  (else else-clause)))

       Eva demonstrates the program for Alyssa:
       (new-if (= 2 3) 0 5)
       (new-if (= 1 1) 0 5)

       Delighted, Alyssa uses new-if to rewrite the square-root
       (define (sqrt-iter guess x)
           (new-if (good-enough? guess x)
                     (sqrt-iter (improve guess x) x)))

       What happens when Alyssa aempts to use this to compute
       square roots? Explain.
  24 Readers who are worried about the efficiency issues involved in using procedure

calls to implement iteration should note the remarks on “tail recursion” in Section 1.2.1.

      Exercise 1.7: e good-enough? test used in computing
      square roots will not be very effective for finding the square
      roots of very small numbers. Also, in real computers, arith-
      metic operations are almost always performed with lim-
      ited precision. is makes our test inadequate for very large
      numbers. Explain these statements, with examples showing
      how the test fails for small and large numbers. An alterna-
      tive strategy for implementing good-enough? is to watch
      how guess changes from one iteration to the next and to
      stop when the change is a very small fraction of the guess.
      Design a square-root procedure that uses this kind of end
      test. Does this work beer for small and large numbers?

      Exercise 1.8: Newton’s method for cube roots is based on
      the fact that if y is an approximation to the cube root of x,
      then a beer approximation is given by the value

                               x/y 2 + 2y
      Use this formula to implement a cube-root procedure anal-
      ogous to the square-root procedure. (In Section 1.3.4 we will
      see how to implement Newton’s method in general as an
      abstraction of these square-root and cube-root procedures.)

1.1.8 Procedures as Black-Box Abstractions
sqrt is our first example of a process defined by a set of mutually defined
procedures. Notice that the definition of sqrt-iter is recursive; that is,
the procedure is defined in terms of itself. e idea of being able to
define a procedure in terms of itself may be disturbing; it may seem

                                /        \
                        good-enough     improve
                          /     \           \
                      square    abs       average

      Figure 1.2: Procedural decomposition of the sqrt program.

unclear how such a “circular” definition could make sense at all, much
less specify a well-defined process to be carried out by a computer. is
will be addressed more carefully in Section 1.2. But first let’s consider
some other important points illustrated by the sqrt example.
    Observe that the problem of computing square roots breaks up nat-
urally into a number of subproblems: how to tell whether a guess is
good enough, how to improve a guess, and so on. Each of these tasks is
accomplished by a separate procedure. e entire sqrt program can be
viewed as a cluster of procedures (shown in Figure 1.2) that mirrors the
decomposition of the problem into subproblems.
    e importance of this decomposition strategy is not simply that
one is dividing the program into parts. Aer all, we could take any large
program and divide it into parts—the first ten lines, the next ten lines,
the next ten lines, and so on. Rather, it is crucial that each procedure ac-
complishes an identifiable task that can be used as a module in defining
other procedures. For example, when we define the good-enough? pro-
cedure in terms of square, we are able to regard the square procedure
as a “black box.” We are not at that moment concerned with how the
procedure computes its result, only with the fact that it computes the
square. e details of how the square is computed can be suppressed,
to be considered at a later time. Indeed, as far as the good-enough? pro-

cedure is concerned, square is not quite a procedure but rather an ab-
straction of a procedure, a so-called procedural abstraction. At this level
of abstraction, any procedure that computes the square is equally good.
    us, considering only the values they return, the following two
procedures for squaring a number should be indistinguishable. Each
takes a numerical argument and produces the square of that number
as the value.25
(define (square x) (* x x))
(define (square x) (exp (double (log x))))
(define (double x) (+ x x))

So a procedure definition should be able to suppress detail. e users
of the procedure may not have wrien the procedure themselves, but
may have obtained it from another programmer as a black box. A user
should not need to know how the procedure is implemented in order to
use it.

Local names
One detail of a procedure’s implementation that should not maer to
the user of the procedure is the implementer’s choice of names for the
procedure’s formal parameters. us, the following procedures should
not be distinguishable:
(define (square x) (* x x))
(define (square y) (* y y))

  25 Itis not even clear which of these procedures is a more efficient implementation.
is depends upon the hardware available. ere are machines for which the “obvious”
implementation is the less efficient one. Consider a machine that has extensive tables
of logarithms and antilogarithms stored in a very efficient manner.

is principle—that the meaning of a procedure should be independent
of the parameter names used by its author—seems on the surface to
be self-evident, but its consequences are profound. e simplest conse-
quence is that the parameter names of a procedure must be local to the
body of the procedure. For example, we used square in the definition
of good-enough? in our square-root procedure:
(define (good-enough? guess x)
  (< (abs (- (square guess) x))

e intention of the author of good-enough? is to determine if the square
of the first argument is within a given tolerance of the second argument.
We see that the author of good-enough? used the name guess to refer to
the first argument and x to refer to the second argument. e argument
of square is guess. If the author of square used x (as above) to refer to
that argument, we see that the x in good-enough? must be a different x
than the one in square. Running the procedure square must not affect
the value of x that is used by good-enough?, because that value of x
may be needed by good-enough? aer square is done computing.
    If the parameters were not local to the bodies of their respective
procedures, then the parameter x in square could be confused with the
parameter x in good-enough?, and the behavior of good-enough? would
depend upon which version of square we used. us, square would not
be the black box we desired.
    A formal parameter of a procedure has a very special role in the
procedure definition, in that it doesn’t maer what name the formal
parameter has. Such a name is called a bound variable, and we say that
the procedure definition binds its formal parameters. e meaning of
a procedure definition is unchanged if a bound variable is consistently

renamed throughout the definition.26 If a variable is not bound, we say
that it is free. e set of expressions for which a binding defines a name
is called the scope of that name. In a procedure definition, the bound
variables declared as the formal parameters of the procedure have the
body of the procedure as their scope.
     In the definition of good-enough? above, guess and x are bound
variables but <, -, abs, and square are free. e meaning of good-
enough? should be independent of the names we choose for guess and
x so long as they are distinct and different from <, -, abs, and square.
(If we renamed guess to abs we would have introduced a bug by cap-
turing the variable abs. It would have changed from free to bound.) e
meaning of good-enough? is not independent of the names of its free
variables, however. It surely depends upon the fact (external to this def-
inition) that the symbol abs names a procedure for computing the abso-
lute value of a number. good-enough? will compute a different function
if we substitute cos for abs in its definition.

Internal definitions and block structure
We have one kind of name isolation available to us so far: e formal
parameters of a procedure are local to the body of the procedure. e
square-root program illustrates another way in which we would like
to control the use of names. e existing program consists of separate
(define (sqrt x)
  (sqrt-iter 1.0 x))
(define (sqrt-iter guess x)
  (if (good-enough? guess x)

 26 e concept of consistent renaming is actually subtle and difficult to define for-

mally. Famous logicians have made embarrassing errors here.

      (sqrt-iter (improve guess x) x)))
(define (good-enough? guess x)
  (< (abs (- (square guess) x)) 0.001))
(define (improve guess x)
  (average guess (/ x guess)))

e problem with this program is that the only procedure that is impor-
tant to users of sqrt is sqrt. e other procedures (sqrt-iter, good-
enough?, and improve) only cluer up their minds. ey may not define
any other procedure called good-enough? as part of another program
to work together with the square-root program, because sqrt needs it.
e problem is especially severe in the construction of large systems
by many separate programmers. For example, in the construction of a
large library of numerical procedures, many numerical functions are
computed as successive approximations and thus might have proce-
dures named good-enough? and improve as auxiliary procedures. We
would like to localize the subprocedures, hiding them inside sqrt so
that sqrt could coexist with other successive approximations, each hav-
ing its own private good-enough? procedure. To make this possible, we
allow a procedure to have internal definitions that are local to that pro-
cedure. For example, in the square-root problem we can write
(define (sqrt x)
  (define (good-enough? guess x)
    (< (abs (- (square guess) x)) 0.001))
  (define (improve guess x) (average guess (/ x guess)))
  (define (sqrt-iter guess x)
    (if (good-enough? guess x)
         (sqrt-iter (improve guess x) x)))
  (sqrt-iter 1.0 x))

Such nesting of definitions, called block structure, is basically the right
solution to the simplest name-packaging problem. But there is a bet-
ter idea lurking here. In addition to internalizing the definitions of the
auxiliary procedures, we can simplify them. Since x is bound in the defi-
nition of sqrt, the procedures good-enough?, improve, and sqrt-iter,
which are defined internally to sqrt, are in the scope of x. us, it is
not necessary to pass x explicitly to each of these procedures. Instead,
we allow x to be a free variable in the internal definitions, as shown be-
low. en x gets its value from the argument with which the enclosing
procedure sqrt is called. is discipline is called lexical scoping.27
(define (sqrt x)
  (define (good-enough? guess)
     (< (abs (- (square guess) x)) 0.001))
  (define (improve guess)
     (average guess (/ x guess)))
  (define (sqrt-iter guess)
     (if (good-enough? guess)
           (sqrt-iter (improve guess))))
  (sqrt-iter 1.0))

We will use block structure extensively to help us break up large pro-
grams into tractable pieces.28 e idea of block structure originated with
the programming language Algol 60. It appears in most advanced pro-
gramming languages and is an important tool for helping to organize
the construction of large programs.
  27 Lexical scoping dictates that free variables in a procedure are taken to refer to
bindings made by enclosing procedure definitions; that is, they are looked up in the
environment in which the procedure was defined. We will see how this works in detail
in chapter 3 when we study environments and the detailed behavior of the interpreter.
   28 Embedded definitions must come first in a procedure body. e management is not

responsible for the consequences of running programs that intertwine definition and
1.2 Procedures and the Processes They Generate
We have now considered the elements of programming: We have used
primitive arithmetic operations, we have combined these operations,
and we have abstracted these composite operations by defining them as
compound procedures. But that is not enough to enable us to say that
we know how to program. Our situation is analogous to that of someone
who has learned the rules for how the pieces move in chess but knows
nothing of typical openings, tactics, or strategy. Like the novice chess
player, we don’t yet know the common paerns of usage in the do-
main. We lack the knowledge of which moves are worth making (which
procedures are worth defining). We lack the experience to predict the
consequences of making a move (executing a procedure).
    e ability to visualize the consequences of the actions under con-
sideration is crucial to becoming an expert programmer, just as it is in
any synthetic, creative activity. In becoming an expert photographer,
for example, one must learn how to look at a scene and know how dark
each region will appear on a print for each possible choice of exposure
and development conditions. Only then can one reason backward, plan-
ning framing, lighting, exposure, and development to obtain the desired
effects. So it is with programming, where we are planning the course
of action to be taken by a process and where we control the process by
means of a program. To become experts, we must learn to visualize the
processes generated by various types of procedures. Only aer we have
developed such a skill can we learn to reliably construct programs that
exhibit the desired behavior.
    A procedure is a paern for the local evolution of a computational
process. It specifies how each stage of the process is built upon the previ-
ous stage. We would like to be able to make statements about the overall,

or global, behavior of a process whose local evolution has been specified
by a procedure. is is very difficult to do in general, but we can at least
try to describe some typical paerns of process evolution.
    In this section we will examine some common “shapes” for pro-
cesses generated by simple procedures. We will also investigate the
rates at which these processes consume the important computational
resources of time and space. e procedures we will consider are very
simple. eir role is like that played by test paerns in photography: as
oversimplified prototypical paerns, rather than practical examples in
their own right.

1.2.1 Linear Recursion and Iteration
We begin by considering the factorial function, defined by

                    n! = n · (n − 1) · (n − 2) · · · 3 · 2 · 1.

ere are many ways to compute factorials. One way is to make use
of the observation that n! is equal to n times (n − 1)! for any positive
integer n:

           n! = n · [(n − 1) · (n − 2) · · · 3 · 2 · 1] = n · (n − 1)!.

us, we can compute n! by computing (n − 1)! and multiplying the
result by n. If we add the stipulation that 1! is equal to 1, this observation
translates directly into a procedure:
(define (factorial n)
  (if (= n 1)
       (* n (factorial (- n 1)))))

          (factorial 6)
          (* 6 (factorial 5))
          (* 6   (* 5   (factorial 4)))
          (* 6   (* 5   (* 4 (factorial 3))))
          (* 6   (* 5   (* 4 (* 3 (factorial 2)))))
          (* 6   (* 5   (* 4 (* 3 (* 2 (factorial 1))))))
          (* 6   (* 5   (* 4 (* 3 (* 2 1)))))
          (* 6   (* 5   (* 4 (* 3 2))))
          (* 6   (* 5   (* 4 6)))
          (* 6   (* 5   24))
          (* 6   120)

        Figure 1.3: A linear recursive process for computing 6!.

We can use the substitution model of Section 1.1.5 to watch this proce-
dure in action computing 6!, as shown in Figure 1.3.
    Now let’s take a different perspective on computing factorials. We
could describe a rule for computing n! by specifying that we first mul-
tiply 1 by 2, then multiply the result by 3, then by 4, and so on until we
reach n. More formally, we maintain a running product, together with
a counter that counts from 1 up to n. We can describe the computation
by saying that the counter and the product simultaneously change from
one step to the next according to the rule
product ← counter * product
counter ← counter + 1

and stipulating that n! is the value of the product when the counter
exceeds n.
    Once again, we can recast our description as a procedure for com-
puting factorials:29
  29 In a real program we would probably use the block structure introduced in the last

section to hide the definition of fact-iter:
                            (factorial    6)
                            (fact-iter    1 1 6)
                            (fact-iter    1 2 6)
                            (fact-iter    2 3 6)
                            (fact-iter    6 4 6)
                            (fact-iter    24 5 6)
                            (fact-iter    120 6 6)
                            (fact-iter    720 7 6)

         Figure 1.4: A linear iterative process for computing 6!.

(define (factorial n)
  (fact-iter 1 1 n))
(define (fact-iter product counter max-count)
  (if (> counter max-count)
       (fact-iter (* counter product)
                      (+ counter 1)

As before, we can use the substitution model to visualize the process of
computing 6!, as shown in Figure 1.4.
(define (factorial n)
  (define (iter product counter)
    (if (> counter n)
         (iter (* counter product)
                (+ counter 1))))
  (iter 1 1))

  We avoided doing this here so as to minimize the number of things to think about at

     Compare the two processes. From one point of view, they seem
hardly different at all. Both compute the same mathematical function on
the same domain, and each requires a number of steps proportional to n
to compute n!. Indeed, both processes even carry out the same sequence
of multiplications, obtaining the same sequence of partial products. On
the other hand, when we consider the “shapes” of the two processes, we
find that they evolve quite differently.
     Consider the first process. e substitution model reveals a shape of
expansion followed by contraction, indicated by the arrow in Figure 1.3.
e expansion occurs as the process builds up a chain of deferred oper-
ations (in this case, a chain of multiplications). e contraction occurs
as the operations are actually performed. is type of process, charac-
terized by a chain of deferred operations, is called a recursive process.
Carrying out this process requires that the interpreter keep track of the
operations to be performed later on. In the computation of n!, the length
of the chain of deferred multiplications, and hence the amount of infor-
mation needed to keep track of it, grows linearly with n (is proportional
to n), just like the number of steps. Such a process is called a linear re-
cursive process.
     By contrast, the second process does not grow and shrink. At each
step, all we need to keep track of, for any n, are the current values of
the variables product, counter, and max-count. We call this an iterative
process. In general, an iterative process is one whose state can be sum-
marized by a fixed number of state variables, together with a fixed rule
that describes how the state variables should be updated as the process
moves from state to state and an (optional) end test that specifies con-
ditions under which the process should terminate. In computing n!, the
number of steps required grows linearly with n. Such a process is called
a linear iterative process.

     e contrast between the two processes can be seen in another way.
In the iterative case, the program variables provide a complete descrip-
tion of the state of the process at any point. If we stopped the compu-
tation between steps, all we would need to do to resume the computa-
tion is to supply the interpreter with the values of the three program
variables. Not so with the recursive process. In this case there is some
additional “hidden” information, maintained by the interpreter and not
contained in the program variables, which indicates “where the process
is” in negotiating the chain of deferred operations. e longer the chain,
the more information must be maintained.30
     In contrasting iteration and recursion, we must be careful not to
confuse the notion of a recursive process with the notion of a recursive
procedure. When we describe a procedure as recursive, we are referring
to the syntactic fact that the procedure definition refers (either directly
or indirectly) to the procedure itself. But when we describe a process
as following a paern that is, say, linearly recursive, we are speaking
about how the process evolves, not about the syntax of how a procedure
is wrien. It may seem disturbing that we refer to a recursive procedure
such as fact-iter as generating an iterative process. However, the pro-
cess really is iterative: Its state is captured completely by its three state
variables, and an interpreter need keep track of only three variables in
order to execute the process.
     One reason that the distinction between process and procedure may
be confusing is that most implementations of common languages (in-
cluding Ada, Pascal, and C) are designed in such a way that the interpre-
tation of any recursive procedure consumes an amount of memory that
  30 When we discuss the implementation of procedures on register machines in Chap-

ter 5, we will see that any iterative process can be realized “in hardware” as a machine
that has a fixed set of registers and no auxiliary memory. In contrast, realizing a re-
cursive process requires a machine that uses an auxiliary data structure known as a
grows with the number of procedure calls, even when the process de-
scribed is, in principle, iterative. As a consequence, these languages can
describe iterative processes only by resorting to special-purpose “loop-
ing constructs” such as do, repeat, until, for, and while. e imple-
mentation of Scheme we shall consider in Chapter 5 does not share this
defect. It will execute an iterative process in constant space, even if the
iterative process is described by a recursive procedure. An implemen-
tation with this property is called tail-recursive. With a tail-recursive
implementation, iteration can be expressed using the ordinary proce-
dure call mechanism, so that special iteration constructs are useful only
as syntactic sugar.31

        Exercise 1.9: Each of the following two procedures defines
        a method for adding two positive integers in terms of the
        procedures inc, which increments its argument by 1, and
        dec, which decrements its argument by 1.

        (define (+ a b)
            (if (= a 0) b (inc (+ (dec a) b))))
        (define (+ a b)
            (if (= a 0) b (+ (dec a) (inc b))))

        Using the substitution model, illustrate the process gener-
        ated by each procedure in evaluating (+ 4 5). Are these
        processes iterative or recursive?
  31 Tail recursion has long been known as a compiler optimization trick. A coherent
semantic basis for tail recursion was provided by Carl Hewi (1977), who explained it in
terms of the “message-passing” model of computation that we shall discuss in Chapter
3. Inspired by this, Gerald Jay Sussman and Guy Lewis Steele Jr. (see Steele and Sussman
1975) constructed a tail-recursive interpreter for Scheme. Steele later showed how tail
recursion is a consequence of the natural way to compile procedure calls (Steele 1977).
e  standard for Scheme requires that Scheme implementations be tail-recursive.

     Exercise 1.10: e following procedure computes a math-
     ematical function called Ackermann’s function.
     (define (A x y)
       (cond ((= y 0) 0)
                ((= x 0) (* 2 y))
                ((= y 1) 2)
                (else (A (- x 1) (A x (- y 1))))))

     What are the values of the following expressions?
     (A 1 10)
     (A 2 4)
     (A 3 3)

     Consider the following procedures, where A is the proce-
     dure defined above:
     (define (f n) (A 0 n))
     (define (g n) (A 1 n))
     (define (h n) (A 2 n))
     (define (k n) (* 5 n n))

     Give concise mathematical definitions for the functions com-
     puted by the procedures f, g, and h for positive integer val-
     ues of n. For example, (k n) computes 5n 2 .

1.2.2 Tree Recursion
Another common paern of computation is called tree recursion. As an
example, consider computing the sequence of Fibonacci numbers, in
which each number is the sum of the preceding two:

                   0, 1, 1, 2, 3, 5, 8, 13, 21, . . . .

In general, the Fibonacci numbers can be defined by the rule
                      0                       if n = 0,
            Fib(n) = 
                      1                       if n = 1,
                      Fib(n − 1) + Fib(n − 2) otherwise.
We can immediately translate this definition into a recursive procedure
for computing Fibonacci numbers:
(define (fib n)
  (cond ((= n 0) 0)
         ((= n 1) 1)
         (else (+ (fib (- n 1))
                   (fib (- n 2))))))

Consider the paern of this computation. To compute (fib 5), we com-
pute (fib 4) and (fib 3). To compute (fib 4), we compute (fib 3)
and (fib 2). In general, the evolved process looks like a tree, as shown
in Figure 1.5. Notice that the branches split into two at each level (ex-
cept at the boom); this reflects the fact that the fib procedure calls
itself twice each time it is invoked.
     is procedure is instructive as a prototypical tree recursion, but it
is a terrible way to compute Fibonacci numbers because it does so much
redundant computation. Notice in Figure 1.5 that the entire computation
of (fib 3)—almost half the work—is duplicated. In fact, it is not hard to
show that the number of times the procedure will compute (fib 1) or
(fib 0) (the number of leaves in the above tree, in general) is precisely
Fib(n + 1). To get an idea of how bad this is, one can show that the value
of Fib(n) grows exponentially with√n. More precisely (see Exercise 1.13),
Fib(n) is the closest integer to ϕ n / 5, where
                                1+ 5
                           ϕ=            ≈ 1.6180

                                                      fib 5

                              fib 4                                       fib 3

                 fib 3                    fib 2                   fib 2           fib 1

             fib 2       fib 1   fib 1       fib 0        fib 1       fib 0

                          1           1           0           1           0
       fib 1    fib 0

         1           0

      Figure 1.5: e tree-recursive process generated in com-
      puting (fib 5).

is the golden ratio, which satisfies the equation

                                      ϕ 2 = ϕ + 1.

us, the process uses a number of steps that grows exponentially with
the input. On the other hand, the space required grows only linearly
with the input, because we need keep track only of which nodes are
above us in the tree at any point in the computation. In general, the
number of steps required by a tree-recursive process will be propor-
tional to the number of nodes in the tree, while the space required will
be proportional to the maximum depth of the tree.
    We can also formulate an iterative process for computing the Fi-
bonacci numbers. e idea is to use a pair of integers a and b, initialized
to Fib(1) = 1 and Fib(0) = 0, and to repeatedly apply the simultaneous

                                    a ← a + b,
                                    b ← a.
It is not hard to show that, aer applying this transformation n times, a
and b will be equal, respectively, to Fib(n + 1) and Fib(n). us, we can
compute Fibonacci numbers iteratively using the procedure
(define (fib n)
  (fib-iter 1 0 n))
(define (fib-iter a b count)
  (if (= count 0)
          (fib-iter (+ a b) a (- count 1))))

is second method for computing Fib(n) is a linear iteration. e differ-
ence in number of steps required by the two methods—one linear in n,
one growing as fast as Fib(n) itself—is enormous, even for small inputs.
    One should not conclude from this that tree-recursive processes
are useless. When we consider processes that operate on hierarchically
structured data rather than numbers, we will find that tree recursion is
a natural and powerful tool.32 But even in numerical operations, tree-
recursive processes can be useful in helping us to understand and de-
sign programs. For instance, although the first fib procedure is much
less efficient than the second one, it is more straightforward, being lile
more than a translation into Lisp of the definition of the Fibonacci se-
quence. To formulate the iterative algorithm required noticing that the
computation could be recast as an iteration with three state variables.

  32 An example of this was hinted at in Section 1.1.3. e interpreter itself evaluates
expressions using a tree-recursive process.

Example: Counting change
It takes only a bit of cleverness to come up with the iterative Fibonacci
algorithm. In contrast, consider the following problem: How many dif-
ferent ways can we make change of $1.00, given half-dollars, quarters,
dimes, nickels, and pennies? More generally, can we write a procedure
to compute the number of ways to change any given amount of money?
     is problem has a simple solution as a recursive procedure. Sup-
pose we think of the types of coins available as arranged in some order.
en the following relation holds:
     e number of ways to change amount a using n kinds of coins

    • the number of ways to change amount a using all but the first
      kind of coin, plus

    • the number of ways to change amount a − d using all n kinds of
      coins, where d is the denomination of the first kind of coin.

To see why this is true, observe that the ways to make change can be
divided into two groups: those that do not use any of the first kind of
coin, and those that do. erefore, the total number of ways to make
change for some amount is equal to the number of ways to make change
for the amount without using any of the first kind of coin, plus the
number of ways to make change assuming that we do use the first kind
of coin. But the laer number is equal to the number of ways to make
change for the amount that remains aer using a coin of the first kind.
    us, we can recursively reduce the problem of changing a given
amount to the problem of changing smaller amounts using fewer kinds
of coins. Consider this reduction rule carefully, and convince yourself

that we can use it to describe an algorithm if we specify the following
degenerate cases:33
    • If a is exactly 0, we should count that as 1 way to make change.
    • If a is less than 0, we should count that as 0 ways to make change.
    • If n is 0, we should count that as 0 ways to make change.
We can easily translate this description into a recursive procedure:
(define (count-change amount) (cc amount 5))
(define (cc amount kinds-of-coins)
  (cond ((= amount 0) 1)
          ((or (< amount 0) (= kinds-of-coins 0)) 0)
          (else (+ (cc amount
                            (- kinds-of-coins 1))
                      (cc (- amount
(define (first-denomination kinds-of-coins)
  (cond ((= kinds-of-coins 1) 1)
          ((= kinds-of-coins 2) 5)
          ((= kinds-of-coins 3) 10)
          ((= kinds-of-coins 4) 25)
          ((= kinds-of-coins 5) 50)))

(e first-denomination procedure takes as input the number of kinds
of coins available and returns the denomination of the first kind. Here
we are thinking of the coins as arranged in order from largest to small-
est, but any order would do as well.) We can now answer our original
question about changing a dollar:
   33 For example, work through in detail how the reduction rule applies to the problem

of making change for 10 cents using pennies and nickels.

(count-change 100)

count-change generates a tree-recursive process with redundancies sim-
ilar to those in our first implementation of fib. (It will take quite a while
for that 292 to be computed.) On the other hand, it is not obvious how
to design a beer algorithm for computing the result, and we leave this
problem as a challenge. e observation that a tree-recursive process
may be highly inefficient but oen easy to specify and understand has
led people to propose that one could get the best of both worlds by
designing a “smart compiler” that could transform tree-recursive pro-
cedures into more efficient procedures that compute the same result.34

       Exercise 1.11: A function f is defined by the rule that
                 n if n < 3,
        f (n) = 
                 f (n − 1) + 2f (n − 2) + 3f (n − 3) if n ≥ 3.

       Write a procedure that computes f by means of a recursive
       process. Write a procedure that computes f by means of an
       iterative process.

       Exercise 1.12: e following paern of numbers is called
       Pascal’s triangle.
  34 One  approach to coping with redundant computations is to arrange maers so
that we automatically construct a table of values as they are computed. Each time we
are asked to apply the procedure to some argument, we first look to see if the value
is already stored in the table, in which case we avoid performing the redundant com-
putation. is strategy, known as tabulation or memoization, can be implemented in a
straightforward way. Tabulation can sometimes be used to transform processes that
require an exponential number of steps (such as count-change) into processes whose
space and time requirements grow linearly with the input. See Exercise 3.27.

                   1       1
               1       2       1
           1       3       3       1
       1       4       6       4       1
                   . . .

       e numbers at the edge of the triangle are all 1, and each
       number inside the triangle is the sum of the two numbers
       above it.35 Write a procedure that computes elements of
       Pascal’s triangle by means of a recursive process.

           √      1.13: Prove that
                                 √ Fib(n) is the closest integer√ to
       ϕ / 5, where ϕ = (1 + 5)/2. Hint: Let ψ = (1 − 5)/2.

       Use induction and the definition of the Fibonacci numbers√
       (see Section 1.2.2) to prove that Fib(n) = (ϕ n − ψ n )/ 5.

1.2.3 Orders of Growth
e previous examples illustrate that processes can differ considerably
in the rates at which they consume computational resources. One con-
venient way to describe this difference is to use the notion of order of
growth to obtain a gross measure of the resources required by a process
as the inputs become larger.
   35 e elements of Pascal’s triangle are called the binomial coefficients, because the
n throw consists of the coefficients of the terms in the expansion of (x + y)n . is pat-
tern for computing the coefficients appeared in Blaise Pascal’s 1653 seminal work on
probability theory, Traité du triangle arithmétique. According to Knuth (1973), the same
paern appears in the Szu-yuen Yü-chien (“e Precious Mirror of the Four Elements”),
published by the Chinese mathematician Chu Shih-chieh in 1303, in the works of the
twelh-century Persian poet and mathematician Omar Khayyam, and in the works of
the twelh-century Hindu mathematician Bháscara Áchárya.

     Let n be a parameter that measures the size of the problem, and let
R(n) be the amount of resources the process requires for a problem of
size n. In our previous examples we took n to be the number for which
a given function is to be computed, but there are other possibilities. For
instance, if our goal is to compute an approximation to the square root of
a number, we might take n to be the number of digits accuracy required.
For matrix multiplication we might take n to be the number of rows in
the matrices. In general there are a number of properties of the problem
with respect to which it will be desirable to analyze a given process.
Similarly, R(n) might measure the number of internal storage registers
used, the number of elementary machine operations performed, and so
on. In computers that do only a fixed number of operations at a time, the
time required will be proportional to the number of elementary machine
operations performed.
     We say that R(n) has order of growth Θ(f (n)), wrien R(n) = Θ(f (n))
(pronounced “theta of f (n)”), if there are positive constants k 1 and k 2
independent of n such that k 1 f (n) ≤ R(n) ≤ k 2 f (n) for any sufficiently
large value of n. (In other words, for large n, the value R(n) is sandwiched
between k 1 f (n) and k 2 f (n).)
     For instance, with the linear recursive process for computing facto-
rial described in Section 1.2.1 the number of steps grows proportionally
to the input n. us, the steps required for this process grows as Θ(n).
We also saw that the space required grows as Θ(n). For the iterative
factorial, the number of steps is still Θ(n) but the space is Θ(1)—that
is, constant.36 e tree-recursive Fibonacci computation requires Θ(ϕ n )
   36 ese statements mask a great deal of oversimplification. For instance, if we count

process steps as “machine operations” we are making the assumption that the number
of machine operations needed to perform, say, a multiplication is independent of the
size of the numbers to be multiplied, which is false if the numbers are sufficiently large.
Similar remarks hold for the estimates of space. Like the design and description of a
process, the analysis of a process can be carried out at various levels of abstraction.
steps and space Θ(n), where ϕ is the golden ratio described in Section
    Orders of growth provide only a crude description of the behavior
of a process. For example, a process requiring n 2 steps and a process
requiring 1000n 2 steps and a process requiring 3n 2 + 10n + 17 steps all
have Θ(n 2 ) order of growth. On the other hand, order of growth provides
a useful indication of how we may expect the behavior of the process to
change as we change the size of the problem. For a Θ(n) (linear) process,
doubling the size will roughly double the amount of resources used. For
an exponential process, each increment in problem size will multiply the
resource utilization by a constant factor. In the remainder of Section 1.2
we will examine two algorithms whose order of growth is logarithmic,
so that doubling the problem size increases the resource requirement
by a constant amount.

      Exercise 1.14: Draw the tree illustrating the process gen-
      erated by the count-change procedure of Section 1.2.2 in
      making change for 11 cents. What are the orders of growth
      of the space and number of steps used by this process as
      the amount to be changed increases?

      Exercise 1.15: e sine of an angle (specified in radians)
      can be computed by making use of the approximation sin x ≈ x
      if x is sufficiently small, and the trigonometric identity
                                     x         x
                        sin x = 3 sin − 4 sin3
                                     3         3
      to reduce the size of the argument of sin. (For purposes of
      this exercise an angle is considered “sufficiently small” if its
      magnitude is not greater than 0.1 radians.) ese ideas are
      incorporated in the following procedures:

      (define (cube x) (* x x x))
      (define (p x) (- (* 3 x) (* 4 (cube x))))
      (define (sine angle)
          (if (not (> (abs angle) 0.1))
               (p (sine (/ angle 3.0)))))

          a. How many times is the procedure p applied when (sine
             12.15) is evaluated?

          b. What is the order of growth in space and number of
             steps (as a function of a) used by the process generated
             by the sine procedure when (sine a) is evaluated?

1.2.4 Exponentiation
Consider the problem of computing the exponential of a given number.
We would like a procedure that takes as arguments a base b and a posi-
tive integer exponent n and computes b n . One way to do this is via the
recursive definition
                            b n = b · b n −1 ,
                            b 0 = 1,
which translates readily into the procedure
(define (expt b n)
  (if (= n 0)
      (* b (expt b (- n 1)))))

is is a linear recursive process, which requires Θ(n) steps and Θ(n)
space. Just as with factorial, we can readily formulate an equivalent lin-
ear iteration:

(define (expt b n)
  (expt-iter b n 1))
(define (expt-iter b counter product)
  (if (= counter 0)
      (expt-iter b
                  (- counter 1)
                  (* b product))))

is version requires Θ(n) steps and Θ(1) space.
   We can compute exponentials in fewer steps by using successive
squaring. For instance, rather than computing b 8 as

                   b · (b · (b · (b · (b · (b · (b · b)))))) ,

we can compute it using three multiplications:

                                b 2 = b · b,
                                b4 = b2 · b2 ,
                                b8 = b4 · b4.

is method works fine for exponents that are powers of 2. We can
also take advantage of successive squaring in computing exponentials
in general if we use the rule

                      b n = (b n/2 )2         if n is even,
                      b n = b · b n −1        if n is odd.

We can express this method as a procedure:
(define (fast-expt b n)
  (cond ((= n 0) 1)
        ((even? n) (square (fast-expt b (/ n 2))))
        (else (* b (fast-expt b (- n 1))))))

where the predicate to test whether an integer is even is defined in terms
of the primitive procedure remainder by
(define (even? n)
  (= (remainder n 2) 0))

e process evolved by fast-expt grows logarithmically with n in both
space and number of steps. To see this, observe that computing b 2n us-
ing fast-expt requires only one more multiplication than computing
b n . e size of the exponent we can compute therefore doubles (approx-
imately) with every new multiplication we are allowed. us, the num-
ber of multiplications required for an exponent of n grows about as fast
as the logarithm of n to the base 2. e process has Θ(log n) growth.37
      e difference between Θ(log n) growth and Θ(n) growth becomes
striking as n becomes large. For example, fast-expt for n = 1000 re-
quires only 14 multiplications.38 It is also possible to use the idea of
successive squaring to devise an iterative algorithm that computes ex-
ponentials with a logarithmic number of steps (see Exercise 1.16), al-
though, as is oen the case with iterative algorithms, this is not wrien
down so straightforwardly as the recursive algorithm.39

       Exercise 1.16: Design a procedure that evolves an itera-
       tive exponentiation process that uses successive squaring
   37 More precisely, the number of multiplications required is equal to 1 less than the

log base 2 of n plus the number of ones in the binary representation of n. is total
is always less than twice the log base 2 of n. e arbitrary constants k 1 and k 2 in the
definition of order notation imply that, for a logarithmic process, the base to which
logarithms are taken does not maer, so all such processes are described as Θ(log n).
   38 You may wonder why anyone would care about raising numbers to the 1000th

power. See Section 1.2.6.
   39 is iterative algorithm is ancient. It appears in the Chandah-sutra by Áchárya

Pingala, wrien before 200 .. See Knuth 1981, section 4.6.3, for a full discussion and
analysis of this and other methods of exponentiation.

and uses a logarithmic number of steps, as does fast-expt.
(Hint: Using the observation that (b n/2 )2 = (b 2 )n/2 , keep,
along with the exponent n and the base b, an additional
state variable a, and define the state transformation in such
a way that the product ab n is unchanged from state to state.
At the beginning of the process a is taken to be 1, and the
answer is given by the value of a at the end of the process.
In general, the technique of defining an invariant quantity
that remains unchanged from state to state is a powerful
way to think about the design of iterative algorithms.)

Exercise 1.17: e exponentiation algorithms in this sec-
tion are based on performing exponentiation by means of
repeated multiplication. In a similar way, one can perform
integer multiplication by means of repeated addition. e
following multiplication procedure (in which it is assumed
that our language can only add, not multiply) is analogous
to the expt procedure:
(define (* a b)
  (if (= b 0)
       (+ a (* a (- b 1)))))

is algorithm takes a number of steps that is linear in b.
Now suppose we include, together with addition, opera-
tions double, which doubles an integer, and halve, which
divides an (even) integer by 2. Using these, design a mul-
tiplication procedure analogous to fast-expt that uses a
logarithmic number of steps.

       Exercise 1.18: Using the results of Exercise 1.16 and Exer-
       cise 1.17, devise a procedure that generates an iterative pro-
       cess for multiplying two integers in terms of adding, dou-
       bling, and halving and uses a logarithmic number of steps.40

       Exercise 1.19: ere is a clever algorithm for computing
       the Fibonacci numbers in a logarithmic number of steps.
       Recall the transformation of the state variables a and b in
       the fib-iter process of Section 1.2.2: a ← a +b and b ← a.
       Call this transformation T , and observe that applying T
       over and over again n times, starting with 1 and 0, produces
       the pair Fib(n + 1) and Fib(n). In other words, the Fibonacci
       numbers are produced by applying T n , the n th power of the
       transformationT , starting with the pair (1, 0). Now consider
       T to be the special case of p = 0 and q = 1 in a family of
       transformations Tpq , where Tpq transforms the pair (a, b)
       according to a ← bq + aq + ap and b ← bp + aq. Show
       that if we apply such a transformation Tpq twice, the effect
       is the same as using a single transformation Tp ′q ′ of the
       same form, and compute p ′ and q ′ in terms of p and q. is
       gives us an explicit way to square these transformations,
       and thus we can compute T n using successive squaring, as
       in the fast-expt procedure. Put this all together to com-
       plete the following procedure, which runs in a logarithmic
       number of steps:41
   40 is algorithm, which is sometimes known as the “Russian peasant method” of

multiplication, is ancient. Examples of its use are found in the Rhind Papyrus, one of the
two oldest mathematical documents in existence, wrien about 1700 .. (and copied
from an even older document) by an Egyptian scribe named A’h-mose.
   41 is exercise was suggested to us by Joe Stoy, based on an example in Kaldewaij

      (define (fib n)
        (fib-iter 1 0 0 1 n))
      (define (fib-iter a b p q count)
        (cond ((= count 0) b)
               ((even? count)
                (fib-iter a
                            ⟨??⟩       ; compute p ′
                            ⟨??⟩       ; compute q ′
                            (/ count 2)))
               (else (fib-iter (+ (* b q) (* a q) (* a p))
                                   (+ (* b p) (* a q))
                                   (- count 1)))))

1.2.5 Greatest Common Divisors
e greatest common divisor () of two integers a and b is defined to
be the largest integer that divides both a and b with no remainder. For
example, the  of 16 and 28 is 4. In Chapter 2, when we investigate
how to implement rational-number arithmetic, we will need to be able
to compute s in order to reduce rational numbers to lowest terms.
(To reduce a rational number to lowest terms, we must divide both the
numerator and the denominator by their . For example, 16/28 re-
duces to 4/7.) One way to find the  of two integers is to factor them
and search for common factors, but there is a famous algorithm that is
much more efficient.
    e idea of the algorithm is based on the observation that, if r is the
remainder when a is divided by b, then the common divisors of a and b
are precisely the same as the common divisors of b and r . us, we can

use the equation

GCD(a,b) = GCD(b,r)

to successively reduce the problem of computing a  to the problem
of computing the  of smaller and smaller pairs of integers. For ex-

GCD(206,40) = GCD(40,6)
                = GCD(6,4)
                = GCD(4,2)
                = GCD(2,0)
                = 2

reduces (, ) to (, ), which is 2. It is possible to show that
starting with any two positive integers and performing repeated reduc-
tions will always eventually produce a pair where the second number is
0. en the  is the other number in the pair. is method for com-
puting the  is known as Euclid’s Algorithm.42
    It is easy to express Euclid’s Algorithm as a procedure:
(define (gcd a b)
  (if (= b 0)
        (gcd b (remainder a b))))

is generates an iterative process, whose number of steps grows as the
logarithm of the numbers involved.
  42 Euclid’sAlgorithm is so called because it appears in Euclid’s Elements (Book 7, ca.
300 ..). According to Knuth (1973), it can be considered the oldest known nontrivial
algorithm. e ancient Egyptian method of multiplication (Exercise 1.18) is surely older,
but, as Knuth explains, Euclid’s algorithm is the oldest known to have been presented
as a general algorithm, rather than as a set of illustrative examples.

   e fact that the number of steps required by Euclid’s Algorithm
has logarithmic growth bears an interesting relation to the Fibonacci

        Lamé’s Theorem: If Euclid’s Algorithm requires k steps to
        compute the  of some pair, then the smaller number in
        the pair must be greater than or equal to the k th Fibonacci

We can use this theorem to get an order-of-growth estimate for Euclid’s
Algorithm. Let n be the smaller of the two inputs to the procedure.
                                                           √        If the
process takes k steps, then we must have n ≥ Fib(k) ≈ ϕ k / 5. erefore
the number of steps k grows as the logarithm (to the base ϕ) of n. Hence,
the order of growth is Θ(log n).

   43 is    theorem was proved in 1845 by Gabriel Lamé, a French mathematician and
engineer known chiefly for his contributions to mathematical physics. To prove the
theorem, we consider pairs (a k , b k ), where a k ≥ b k , for which Euclid’s Algorithm
terminates in k steps. e proof is based on the claim that, if (a k +1 , b k +1 ) → (a k , b k ) →
(a k −1 , bk −1 ) are three successive pairs in the reduction process, then we must have
bk +1 ≥ bk + b k −1 . To verify the claim, consider that a reduction step is defined by
applying the transformation a k −1 = bk , bk −1 = remainder of a k divided by b k . e
second equation means that a k = qbk + bk −1 for some positive integer q. And since q
must be at least 1 we have a k = qbk + b k −1 ≥ bk + bk −1 . But in the previous reduction
step we have b k +1 = a k . erefore, bk +1 = a k ≥ b k +bk −1 . is verifies the claim. Now
we can prove the theorem by induction on k, the number of steps that the algorithm
requires to terminate. e result is true for k = 1, since this merely requires that b be at
least as large as Fib(1) = 1. Now, assume that the result is true for all integers less than or
equal to k and establish the result for k + 1. Let (a k +1 , bk +1 ) → (a k , bk ) → (a k −1 , bk −1 )
be successive pairs in the reduction process. By our induction hypotheses, we have
bk −1 ≥ Fib(k − 1) and bk ≥ Fib(k). us, applying the claim we just proved together
with the definition of the Fibonacci numbers gives bk +1 ≥ bk + b k −1 ≥ Fib(k) + Fib(k −
1) = Fib(k + 1), which completes the proof of Lamé’s eorem.

      Exercise 1.20: e process that a procedure generates is
      of course dependent on the rules used by the interpreter.
      As an example, consider the iterative gcd procedure given
      above. Suppose we were to interpret this procedure using
      normal-order evaluation, as discussed in Section 1.1.5. (e
      normal-order-evaluation rule for if is described in Exercise
      1.5.) Using the substitution method (for normal order), illus-
      trate the process generated in evaluating (gcd 206 40) and
      indicate the remainder operations that are actually per-
      formed. How many remainder operations are actually per-
      formed in the normal-order evaluation of (gcd 206 40)?
      In the applicative-order evaluation?

1.2.6 Example: Testing for Primality
is section describes two methods for checking the primality of an in-
teger n, one with order of growth Θ( n), and a “probabilistic” algorithm
with order of growth Θ(log n). e exercises at the end of this section
suggest programming projects based on these algorithms.

Searching for divisors
Since ancient times, mathematicians have been fascinated by problems
concerning prime numbers, and many people have worked on the prob-
lem of determining ways to test if numbers are prime. One way to test
if a number is prime is to find the number’s divisors. e following pro-
gram finds the smallest integral divisor (greater than 1) of a given num-
ber n. It does this in a straightforward way, by testing n for divisibility
by successive integers starting with 2.
(define (smallest-divisor n) (find-divisor n 2))

(define (find-divisor n test-divisor)
  (cond ((> (square test-divisor) n) n)
            ((divides? test-divisor n) test-divisor)
            (else (find-divisor n (+ test-divisor 1)))))
(define (divides? a b) (= (remainder b a) 0))

We can test whether a number is prime as follows: n is prime if and only
if n is its own smallest divisor.
(define (prime? n)
  (= n (smallest-divisor n)))

e end test for find-divisor is based on the fact that if n is not prime it
must have a divisor less than or equal to n.44 is means that the algo-
rithm need only test divisors between 1 and n. Consequently, the num-
ber of steps required to identify n as prime will have order of growth
Θ( n).

The Fermat test
e Θ(log n) primality test is based on a result from number theory
known as Fermat’s Lile eorem.45
  44 If d                                                                             √
          is a divisor of n, then so is n/d. But d and n/d cannot both be greater than n.
   45 Pierre de Fermat (1601-1665) is considered to be the founder of modern number the-

ory. He obtained many important number-theoretic results, but he usually announced
just the results, without providing his proofs. Fermat’s Lile eorem was stated in a
leer he wrote in 1640. e first published proof was given by Euler in 1736 (and an
earlier, identical proof was discovered in the unpublished manuscripts of Leibniz). e
most famous of Fermat’s results—known as Fermat’s Last eorem—was joed down
in 1637 in his copy of the book Arithmetic (by the third-century Greek mathematician
Diophantus) with the remark “I have discovered a truly remarkable proof, but this mar-
gin is too small to contain it.” Finding a proof of Fermat’s Last eorem became one of
the most famous challenges in number theory. A complete solution was finally given
in 1995 by Andrew Wiles of Princeton University.

      Fermat’s Lile Theorem: If n is a prime number and a
      is any positive integer less than n, then a raised to the n th
      power is congruent to a modulo n.

(Two numbers are said to be congruent modulo n if they both have the
same remainder when divided by n. e remainder of a number a when
divided by n is also referred to as the remainder of a modulo n, or simply
as a modulo n.)
    If n is not prime, then, in general, most of the numbers a < n will
not satisfy the above relation. is leads to the following algorithm for
testing primality: Given a number n, pick a random number a < n and
compute the remainder of an modulo n. If the result is not equal to a,
then n is certainly not prime. If it is a, then chances are good that n is
prime. Now pick another random number a and test it with the same
method. If it also satisfies the equation, then we can be even more con-
fident that n is prime. By trying more and more values of a, we can
increase our confidence in the result. is algorithm is known as the
Fermat test.
    To implement the Fermat test, we need a procedure that computes
the exponential of a number modulo another number:
(define (expmod base exp m)
  (cond ((= exp 0) 1)
         ((even? exp)
           (square (expmod base (/ exp 2) m))
           (* base (expmod base (- exp 1) m))

is is very similar to the fast-expt procedure of Section 1.2.4. It uses
successive squaring, so that the number of steps grows logarithmically
with the exponent.46
    e Fermat test is performed by choosing at random a number a be-
tween 1 and n −1 inclusive and checking whether the remainder modulo
n of the n th power of a is equal to a. e random number a is chosen us-
ing the procedure random, which we assume is included as a primitive
in Scheme. random returns a nonnegative integer less than its integer
input. Hence, to obtain a random number between 1 and n − 1, we call
random with an input of n − 1 and add 1 to the result:

(define (fermat-test n)
  (define (try-it a)
     (= (expmod a n n) a))
  (try-it (+ 1 (random (- n 1)))))

e following procedure runs the test a given number of times, as spec-
ified by a parameter. Its value is true if the test succeeds every time, and
false otherwise.
(define (fast-prime? n times)
  (cond ((= times 0) true)
          ((fermat-test n) (fast-prime? n (- times 1)))
          (else false)))

  46 e  reduction steps in the cases where the exponent e is greater than 1 are based
on the fact that, for any integers x , y, and m, we can find the remainder of x times y
modulo m by computing separately the remainders of x modulo m and y modulo m,
multiplying these, and then taking the remainder of the result modulo m. For instance,
in the case where e is even, we compute the remainder of b e/2 modulo m, square this,
and take the remainder modulo m. is technique is useful because it means we can
perform our computation without ever having to deal with numbers much larger than
m. (Compare Exercise 1.25.)

Probabilistic methods
e Fermat test differs in character from most familiar algorithms, in
which one computes an answer that is guaranteed to be correct. Here,
the answer obtained is only probably correct. More precisely, if n ever
fails the Fermat test, we can be certain that n is not prime. But the fact
that n passes the test, while an extremely strong indication, is still not
a guarantee that n is prime. What we would like to say is that for any
number n, if we perform the test enough times and find that n always
passes the test, then the probability of error in our primality test can be
made as small as we like.
     Unfortunately, this assertion is not quite correct. ere do exist num-
bers that fool the Fermat test: numbers n that are not prime and yet have
the property that an is congruent to a modulo n for all integers a < n.
Such numbers are extremely rare, so the Fermat test is quite reliable in
     ere are variations of the Fermat test that cannot be fooled. In these
tests, as with the Fermat method, one tests the primality of an integer n
by choosing a random integer a < n and checking some condition that
depends upon n and a. (See Exercise 1.28 for an example of such a test.)
On the other hand, in contrast to the Fermat test, one can prove that,
for any n, the condition does not hold for most of the integers a < n
unless n is prime. us, if n passes the test for some random choice of
   47 Numbers that fool the Fermat test are called Carmichael numbers, and lile is

known about them other than that they are extremely rare. ere are 255 Carmichael
numbers below 100,000,000. e smallest few are 561, 1105, 1729, 2465, 2821, and 6601.
In testing primality of very large numbers chosen at random, the chance of stumbling
upon a value that fools the Fermat test is less than the chance that cosmic radiation will
cause the computer to make an error in carrying out a “correct” algorithm. Considering
an algorithm to be inadequate for the first reason but not for the second illustrates the
difference between mathematics and engineering.

a, the chances are beer than even that n is prime. If n passes the test
for two random choices of a, the chances are beer than 3 out of 4 that
n is prime. By running the test with more and more randomly chosen
values of a we can make the probability of error as small as we like.
    e existence of tests for which one can prove that the chance of
error becomes arbitrarily small has sparked interest in algorithms of this
type, which have come to be known as probabilistic algorithms. ere is
a great deal of research activity in this area, and probabilistic algorithms
have been fruitfully applied to many fields.48

       Exercise 1.21: Use the smallest-divisor procedure to find
       the smallest divisor of each of the following numbers: 199,
       1999, 19999.

       Exercise 1.22: Most Lisp implementations include a prim-
       itive called runtime that returns an integer that specifies
       the amount of time the system has been running (mea-
       sured, for example, in microseconds). e following timed-
       prime-test procedure, when called with an integer n, prints
       n and checks to see if n is prime. If n is prime, the procedure
       prints three asterisks followed by the amount of time used
       in performing the test.
   48 One of the most striking applications of probabilistic prime testing has been to the

field of cryptography. Although it is now computationally infeasible to factor an arbi-
trary 200-digit number, the primality of such a number can be checked in a few seconds
with the Fermat test. is fact forms the basis of a technique for constructing “unbreak-
able codes” suggested by Rivest et al. (1977). e resulting RSA algorithm has become
a widely used technique for enhancing the security of electronic communications. Be-
cause of this and related developments, the study of prime numbers, once considered
the epitome of a topic in “pure” mathematics to be studied only for its own sake, now
turns out to have important practical applications to cryptography, electronic funds
transfer, and information retrieval.

(define (timed-prime-test n)
  (display n)
  (start-prime-test n (runtime)))
(define (start-prime-test n start-time)
  (if (prime? n)
       (report-prime (- (runtime) start-time))))
(define (report-prime elapsed-time)
  (display " *** ")
  (display elapsed-time))

Using this procedure, write a procedure search-for-primes
that checks the primality of consecutive odd integers in a
specified range. Use your procedure to find the three small-
est primes larger than 1000; larger than 10,000; larger than
100,000; larger than 1,000,000. Note the time needed to test
each prime. Since the testing algorithm has order of growth
of Θ( n), you should expect√ that testing for primes around
10,000 should take about 10 times as long as testing for
primes around 1000. Do your timing data bear this out?
How well do the data for 100,000 and 1,000,000 support the
Θ( n) prediction? Is your result compatible with the notion
that programs on your machine run in time proportional to
the number of steps required for the computation?

Exercise 1.23: e smallest-divisor procedure shown at
the start of this section does lots of needless testing: Aer it
checks to see if the number is divisible by 2 there is no point
in checking to see if it is divisible by any larger even num-
bers. is suggests that the values used for test-divisor
should not be 2, 3, 4, 5, 6, . . ., but rather 2, 3, 5, 7, 9, . . ..

To implement this change, define a procedure next that re-
turns 3 if its input is equal to 2 and otherwise returns its in-
put plus 2. Modify the smallest-divisor procedure to use
(next test-divisor) instead of (+ test-divisor 1).
With timed-prime-test incorporating this modified ver-
sion of smallest-divisor, run the test for each of the 12
primes found in Exercise 1.22. Since this modification halves
the number of test steps, you should expect it to run about
twice as fast. Is this expectation confirmed? If not, what is
the observed ratio of the speeds of the two algorithms, and
how do you explain the fact that it is different from 2?

Exercise 1.24: Modify the timed-prime-test procedure of
Exercise 1.22 to use fast-prime? (the Fermat method), and
test each of the 12 primes you found in that exercise. Since
the Fermat test has Θ(log n) growth, how would you expect
the time to test primes near 1,000,000 to compare with the
time needed to test primes near 1000? Do your data bear
this out? Can you explain any discrepancy you find?

Exercise 1.25: Alyssa P. Hacker complains that we went to
a lot of extra work in writing expmod. Aer all, she says,
since we already know how to compute exponentials, we
could have simply wrien
(define (expmod base exp m)
  (remainder (fast-expt base exp) m))

Is she correct? Would this procedure serve as well for our
fast prime tester? Explain.

Exercise 1.26: Louis Reasoner is having great difficulty do-
ing Exercise 1.24. His fast-prime? test seems to run more
slowly than his prime? test. Louis calls his friend Eva Lu
Ator over to help. When they examine Louis’s code, they
find that he has rewrien the expmod procedure to use an
explicit multiplication, rather than calling square:
(define (expmod base exp m)
  (cond ((= exp 0) 1)
         ((even? exp)
          (remainder (* (expmod base (/ exp 2) m)
                           (expmod base (/ exp 2) m))
          (remainder (* base
                           (expmod base (- exp 1) m))

“I don’t see what difference that could make,” says Louis.
“I do.” says Eva. “By writing the procedure like that, you
have transformed the Θ(log n) process into a Θ(n) process.”

Exercise 1.27: Demonstrate that the Carmichael numbers
listed in Footnote 1.47 really do fool the Fermat test. at is,
write a procedure that takes an integer n and tests whether
an is congruent to a modulo n for every a < n, and try your
procedure on the given Carmichael numbers.

Exercise 1.28: One variant of the Fermat test that cannot
be fooled is called the Miller-Rabin test (Miller 1976; Rabin
1980). is starts from an alternate form of Fermat’s Lile

     eorem, which states that if n is a prime number and a is
     any positive integer less than n, then a raised to the (n−1)-st
     power is congruent to 1 modulo n. To test the primality of a
     number n by the Miller-Rabin test, we pick a random num-
     ber a < n and raise a to the (n − 1)-st power modulo n using
     the expmod procedure. However, whenever we perform the
     squaring step in expmod, we check to see if we have discov-
     ered a “nontrivial square root of 1 modulo n,” that is, a num-
     ber not equal to 1 or n −1 whose square is equal to 1 modulo
     n. It is possible to prove that if such a nontrivial square root
     of 1 exists, then n is not prime. It is also possible to prove
     that if n is an odd number that is not prime, then, for at least
     half the numbers a < n, computing an −1 in this way will
     reveal a nontrivial square root of 1 modulo n. (is is why
     the Miller-Rabin test cannot be fooled.) Modify the expmod
     procedure to signal if it discovers a nontrivial square root
     of 1, and use this to implement the Miller-Rabin test with
     a procedure analogous to fermat-test. Check your pro-
     cedure by testing various known primes and non-primes.
     Hint: One convenient way to make expmod signal is to have
     it return 0.

1.3 Formulating Abstractions
    with Higher-Order Procedures
We have seen that procedures are, in effect, abstractions that describe
compound operations on numbers independent of the particular num-
bers. For example, when we

(define (cube x) (* x x x))

we are not talking about the cube of a particular number, but rather
about a method for obtaining the cube of any number. Of course we
could get along without ever defining this procedure, by always writing
expressions such as
(* 3 3 3)
(* x x x)
(* y y y)

and never mentioning cube explicitly. is would place us at a serious
disadvantage, forcing us to work always at the level of the particular op-
erations that happen to be primitives in the language (multiplication, in
this case) rather than in terms of higher-level operations. Our programs
would be able to compute cubes, but our language would lack the ability
to express the concept of cubing. One of the things we should demand
from a powerful programming language is the ability to build abstrac-
tions by assigning names to common paerns and then to work in terms
of the abstractions directly. Procedures provide this ability. is is why
all but the most primitive programming languages include mechanisms
for defining procedures.
     Yet even in numerical processing we will be severely limited in our
ability to create abstractions if we are restricted to procedures whose pa-
rameters must be numbers. Oen the same programming paern will
be used with a number of different procedures. To express such paerns
as concepts, we will need to construct procedures that can accept pro-
cedures as arguments or return procedures as values. Procedures that
manipulate procedures are called higher-order procedures. is section
shows how higher-order procedures can serve as powerful abstraction
mechanisms, vastly increasing the expressive power of our language.

1.3.1 Procedures as Arguments
Consider the following three procedures. e first computes the sum of
the integers from a through b:
(define (sum-integers a b)
  (if (> a b)
        (+ a (sum-integers (+ a 1) b))))

e second computes the sum of the cubes of the integers in the given
(define (sum-cubes a b)
  (if (> a b)
        (+ (cube a)
            (sum-cubes (+ a 1) b))))

e third computes the sum of a sequence of terms in the series
                              1     1      1
                                 +     +       +...,
                            1 · 3 5 · 7 9 · 11
which converges to π /8 (very slowly):49
(define (pi-sum a b)
  (if (> a b)
        (+ (/ 1.0 (* a (+ a 2)))
            (pi-sum (+ a 4) b))))

    is series, usually wrien in the equivalent form π4 = 1 − 13 + 15 − 17 + . . ., is due
to Leibniz. We’ll see how to use this as the basis for some fancy numerical tricks in
Section 3.5.3.

ese three procedures clearly share a common underlying paern.
ey are for the most part identical, differing only in the name of the
procedure, the function of a used to compute the term to be added, and
the function that provides the next value of a. We could generate each
of the procedures by filling in slots in the same template:
(define (⟨name⟩ a b)
  (if (> a b)
      (+ (⟨term⟩ a)
          (⟨name⟩ (⟨next⟩ a) b))))

e presence of such a common paern is strong evidence that there is
a useful abstraction waiting to be brought to the surface. Indeed, math-
ematicians long ago identified the abstraction of summation of a series
and invented “sigma notation,” for example
                             f (n) = f (a) + · · · + f (b),

to express this concept. e power of sigma notation is that it allows
mathematicians to deal with the concept of summation itself rather than
only with particular sums—for example, to formulate general results
about sums that are independent of the particular series being summed.
    Similarly, as program designers, we would like our language to be
powerful enough so that we can write a procedure that expresses the
concept of summation itself rather than only procedures that compute
particular sums. We can do so readily in our procedural language by
taking the common template shown above and transforming the “slots”
into formal parameters:
(define (sum term a next b)
  (if (> a b)

       (+ (term a)
           (sum term (next a) next b))))

Notice that sum takes as its arguments the lower and upper bounds a
and b together with the procedures term and next. We can use sum just
as we would any procedure. For example, we can use it (along with a
procedure inc that increments its argument by 1) to define sum-cubes:
(define (inc n) (+ n 1))
(define (sum-cubes a b)
  (sum cube a inc b))

Using this, we can compute the sum of the cubes of the integers from 1
to 10:
(sum-cubes 1 10)

With the aid of an identity procedure to compute the term, we can define
sum-integers in terms of sum:

(define (identity x) x)
(define (sum-integers a b)
  (sum identity a inc b))

en we can add up the integers from 1 to 10:
(sum-integers 1 10)

We can also define pi-sum in the same way:50
  50 Notice that we have used block structure (Section 1.1.8) to embed the definitions of

pi-next  and pi-term within pi-sum, since these procedures are unlikely to be useful
for any other purpose. We will see how to get rid of them altogether in Section 1.3.2.

(define (pi-sum a b)
  (define (pi-term x)
    (/ 1.0 (* x (+ x 2))))
  (define (pi-next x)
    (+ x 4))
  (sum pi-term a pi-next b))

Using these procedures, we can compute an approximation to π :
(* 8 (pi-sum 1 1000))

Once we have sum, we can use it as a building block in formulating fur-
ther concepts. For instance, the definite integral of a function f between
the limits a and b can be approximated numerically using the formula
  ∫ b      [ (        )    (            )     (              )        ]
                   dx                dx                   dx
      f = f a+          + f a + dx +      + f a + 2dx +        + . . . dx
    a               2                 2                    2

for small values of dx. We can express this directly as a procedure:
(define (integral f a b dx)
  (define (add-dx x)
    (+ x dx))
  (* (sum f (+ a (/ dx 2.0)) add-dx b)

(integral cube 0 1 0.01)

(integral cube 0 1 0.001)

(e exact value of the integral of cube between 0 and 1 is 1/4.)

Exercise 1.29: Simpson’s Rule is a more accurate method
of numerical integration than the method illustrated above.
Using Simpson’s Rule, the integral of a function f between
a and b is approximated as
     (y0 + 4y1 + 2y2 + 4y3 + 2y4 + · · · + 2yn −2 + 4yn −1 + yn ),
where h = (b − a)/n, for some even integer n, and yk =
f (a + kh). (Increasing n increases the accuracy of the ap-
proximation.) Define a procedure that takes as arguments
f , a, b, and n and returns the value of the integral, com-
puted using Simpson’s Rule. Use your procedure to inte-
grate cube between 0 and 1 (with n = 100 and n = 1000),
and compare the results to those of the integral procedure
shown above.

Exercise 1.30: e sum procedure above generates a linear
recursion. e procedure can be rewrien so that the sum
is performed iteratively. Show how to do this by filling in
the missing expressions in the following definition:
(define (sum term a next b)
  (define (iter a result)
      (if   ⟨??⟩
            (iter  ⟨??⟩ ⟨??⟩)))
  (iter     ⟨??⟩ ⟨??⟩))

Exercise 1.31:

  a. e sum procedure is only the simplest of a vast num-
     ber of similar abstractions that can be captured as higher-

              order procedures.51 Write an analogous procedure called
              product that returns the product of the values of a
              function at points over a given range. Show how to de-
              fine factorial in terms of product. Also use product
              to compute approximations to π using the formula52
                               π   2 · 4 · 4 · 6 · 6 · 8···
                                 =                          .
                               4 3 · 3 · 5 · 5 · 7 · 7···
          b. If your product procedure generates a recursive pro-
             cess, write one that generates an iterative process. If
             it generates an iterative process, write one that gen-
             erates a recursive process.

       Exercise 1.32:

          a. Show that sum and product (Exercise 1.31) are both
             special cases of a still more general notion called accumulate
             that combines a collection of terms, using some gen-
             eral accumulation function:
              (accumulate combiner null-value term a next b)

   51 e intent of Exercise 1.31 through Exercise 1.33 is to demonstrate the expressive

power that is aained by using an appropriate abstraction to consolidate many seem-
ingly disparate operations. However, though accumulation and filtering are elegant
ideas, our hands are somewhat tied in using them at this point since we do not yet
have data structures to provide suitable means of combination for these abstractions.
We will return to these ideas in Section 2.2.3 when we show how to use sequences as
interfaces for combining filters and accumulators to build even more powerful abstrac-
tions. We will see there how these methods really come into their own as a powerful
and elegant approach to designing programs.
   52 is formula was discovered by the seventeenth-century English mathematician

John Wallis.

     accumulate   takes as arguments the same term and
     range specifications as sum and product, together with
     a combiner procedure (of two arguments) that speci-
     fies how the current term is to be combined with the
     accumulation of the preceding terms and a null-value
     that specifies what base value to use when the terms
     run out. Write accumulate and show how sum and
     product can both be defined as simple calls to accumulate.
  b. If your accumulate procedure generates a recursive
     process, write one that generates an iterative process.
     If it generates an iterative process, write one that gen-
     erates a recursive process.

Exercise 1.33: You can obtain an even more general ver-
sion of accumulate (Exercise 1.32) by introducing the no-
tion of a filter on the terms to be combined. at is, combine
only those terms derived from values in the range that sat-
isfy a specified condition. e resulting filtered-accumulate
abstraction takes the same arguments as accumulate, to-
gether with an additional predicate of one argument that
specifies the filter. Write filtered-accumulate as a proce-
dure. Show how to express the following using filtered-

  a. the sum of the squares of the prime numbers in the
     interval a to b (assuming that you have a prime? pred-
     icate already wrien)
  b. the product of all the positive integers less than n that
     are relatively prime to n (i.e., all positive integers i < n
     such that (i, n) = 1).

1.3.2 Constructing Procedures Using lambda
In using sum as in Section 1.3.1, it seems terribly awkward to have to
define trivial procedures such as pi-term and pi-next just so we can
use them as arguments to our higher-order procedure. Rather than de-
fine pi-next and pi-term, it would be more convenient to have a way
to directly specify “the procedure that returns its input incremented by
4” and “the procedure that returns the reciprocal of its input times its
input plus 2.” We can do this by introducing the special form lambda,
which creates procedures. Using lambda we can describe what we want
(lambda (x) (+ x 4))

(lambda (x) (/ 1.0 (* x (+ x 2))))

en our pi-sum procedure can be expressed without defining any aux-
iliary procedures as
(define (pi-sum a b)
  (sum (lambda (x) (/ 1.0 (* x (+ x 2))))
       (lambda (x) (+ x 4))

Again using lambda, we can write the integral procedure without hav-
ing to define the auxiliary procedure add-dx:
(define (integral f a b dx)
  (* (sum f
             (+ a (/ dx 2.0))
             (lambda (x) (+ x dx))

In general, lambda is used to create procedures in the same way as
define, except that no name is specified for the procedure:

(lambda (⟨formal-parameters⟩)          ⟨body⟩)

e resulting procedure is just as much a procedure as one that is cre-
ated using define. e only difference is that it has not been associated
with any name in the environment. In fact,
(define (plus4 x) (+ x 4))

is equivalent to
(define plus4 (lambda (x) (+ x 4)))

We can read a lambda expression as follows:
(lambda                              (x)         (+   x       4))
     |                                |          |    |       |
the procedure of an argument x that adds x and 4

Like any expression that has a procedure as its value, a lambda expres-
sion can be used as the operator in a combination such as
((lambda (x y z) (+ x y (square z)))
 1 2 3)

or, more generally, in any context where we would normally use a pro-
cedure name.53
   53 It would be clearer and less intimidating to people learning Lisp if a name more

obvious than lambda, such as make-procedure, were used. But the convention is firmly
entrenched. e notation is adopted from the λ-calculus, a mathematical formalism in-
troduced by the mathematical logician Alonzo Church (1941). Church developed the
λ-calculus to provide a rigorous foundation for studying the notions of function and
function application. e λ-calculus has become a basic tool for mathematical investi-
gations of the semantics of programming languages.

Using let to create local variables
Another use of lambda is in creating local variables. We oen need lo-
cal variables in our procedures other than those that have been bound
as formal parameters. For example, suppose we wish to compute the

            f (x , y) = x(1 + xy)2 + y(1 − y) + (1 + xy)(1 − y),

which we could also express as

                                 a = 1 + xy,
                                 b = 1 − y,
                         f (x , y) = xa 2 + yb + ab.

In writing a procedure to compute f , we would like to include as local
variables not only x and y but also the names of intermediate quantities
like a and b. One way to accomplish this is to use an auxiliary procedure
to bind the local variables:
(define (f x y)
  (define (f-helper a b)
    (+ (* x (square a))
       (* y b)
       (* a b)))
  (f-helper (+ 1 (* x y))
             (- 1 y)))

Of course, we could use a lambda expression to specify an anonymous
procedure for binding our local variables. e body of f then becomes
a single call to that procedure:
(define (f x y)
  ((lambda (a b)

      (+ (* x (square a))
          (* y b)
          (* a b)))
   (+ 1 (* x y))
   (- 1 y)))

is construct is so useful that there is a special form called let to make
its use more convenient. Using let, the f procedure could be wrien as
(define (f x y)
  (let ((a (+ 1 (* x y)))
          (b (- 1 y)))
      (+ (* x (square a))
         (* y b)
         (* a b))))

e general form of a let expression is
(let ((⟨var1 ⟩    ⟨exp1 ⟩)
       (⟨var2 ⟩   ⟨exp2 ⟩)
       (⟨varn ⟩   ⟨expn ⟩))
which can be thought of as saying
let⟨var1 ⟩ have the value ⟨exp1 ⟩ and
   ⟨var2 ⟩ have the value ⟨exp2 ⟩ and
   ⟨varn ⟩ have the value ⟨expn ⟩
in ⟨body⟩

e first part of the let expression is a list of name-expression pairs.
When the let is evaluated, each name is associated with the value of
the corresponding expression. e body of the let is evaluated with
these names bound as local variables. e way this happens is that the
let expression is interpreted as an alternate syntax for

((lambda (⟨var1 ⟩   . . . ⟨varn ⟩)
 ⟨exp1 ⟩
 ⟨expn ⟩)

No new mechanism is required in the interpreter in order to provide
local variables. A let expression is simply syntactic sugar for the un-
derlying lambda application.
    We can see from this equivalence that the scope of a variable spec-
ified by a let expression is the body of the let. is implies that:

    • let allows one to bind variables as locally as possible to where
      they are to be used. For example, if the value of x is 5, the value
      of the expression
      (+ (let ((x 3))
               (+ x (* x 10)))

      is 38. Here, the x in the body of the let is 3, so the value of the
      let  expression is 33. On the other hand, the x that is the second
      argument to the outermost + is still 5.

    • e variables’ values are computed outside the let. is maers
      when the expressions that provide the values for the local vari-
      ables depend upon variables having the same names as the local
      variables themselves. For example, if the value of x is 2, the ex-
      (let ((x 3)
               (y (+ x 2)))
        (* x y))

       will have the value 12 because, inside the body of the let, x will
       be 3 and y will be 4 (which is the outer x plus 2).

Sometimes we can use internal definitions to get the same effect as with
let. For example, we could have defined the procedure f above as

(define (f x y)
  (define a (+ 1 (* x y)))
  (define b (- 1 y))
  (+ (* x (square a))
      (* y b)
      (* a b)))

We prefer, however, to use let in situations like this and to use internal
define only for internal procedures.54

       Exercise 1.34: Suppose we define the procedure
       (define (f g) (g 2))

       en we have
       (f square)
       (f (lambda (z) (* z (+ z 1))))

       What happens if we (perversely) ask the interpreter to eval-
       uate the combination (f f)? Explain.
  54 Understanding  internal definitions well enough to be sure a program means what
we intend it to mean requires a more elaborate model of the evaluation process than we
have presented in this chapter. e subtleties do not arise with internal definitions of
procedures, however. We will return to this issue in Section 4.1.6, aer we learn more
about evaluation.

1.3.3 Procedures as General Methods
We introduced compound procedures in Section 1.1.4 as a mechanism
for abstracting paerns of numerical operations so as to make them in-
dependent of the particular numbers involved. With higher-order pro-
cedures, such as the integral procedure of Section 1.3.1, we began to
see a more powerful kind of abstraction: procedures used to express
general methods of computation, independent of the particular func-
tions involved. In this section we discuss two more elaborate examples—
general methods for finding zeros and fixed points of functions—and
show how these methods can be expressed directly as procedures.

Finding roots of equations by the half-interval method
e half-interval method is a simple but powerful technique for finding
roots of an equation f (x) = 0, where f is a continuous function. e
idea is that, if we are given points a and b such that f (a) < 0 < f (b),
then f must have at least one zero between a and b. To locate a zero,
let x be the average of a and b, and compute f (x ). If f (x ) > 0, then
f must have a zero between a and x. If f (x) < 0, then f must have a
zero between x and b. Continuing in this way, we can identify smaller
and smaller intervals on which f must have a zero. When we reach a
point where the interval is small enough, the process stops. Since the
interval of uncertainty is reduced by half at each step of the process, the
number of steps required grows as Θ(log(L/T )), where L is the length
of the original interval and T is the error tolerance (that is, the size of
the interval we will consider “small enough”). Here is a procedure that
implements this strategy:
(define (search f neg-point pos-point)
  (let ((midpoint (average neg-point pos-point)))

     (if (close-enough? neg-point pos-point)
          (let ((test-value (f midpoint)))
             (cond ((positive? test-value)
                      (search f neg-point midpoint))
                     ((negative? test-value)
                      (search f midpoint pos-point))
                     (else midpoint))))))

We assume that we are initially given the function f together with
points at which its values are negative and positive. We first compute
the midpoint of the two given points. Next we check to see if the given
interval is small enough, and if so we simply return the midpoint as our
answer. Otherwise, we compute as a test value the value of f at the mid-
point. If the test value is positive, then we continue the process with a
new interval running from the original negative point to the midpoint.
If the test value is negative, we continue with the interval from the mid-
point to the positive point. Finally, there is the possibility that the test
value is 0, in which case the midpoint is itself the root we are searching
     To test whether the endpoints are “close enough” we can use a pro-
cedure similar to the one used in Section 1.1.7 for computing square
(define (close-enough? x y) (< (abs (- x y)) 0.001))

search  is awkward to use directly, because we can accidentally give it
points at which f ’s values do not have the required sign, in which case
  55 We have used 0.001 as a representative “small” number to indicate a tolerance for
the acceptable error in a calculation. e appropriate tolerance for a real calculation
depends upon the problem to be solved and the limitations of the computer and the
algorithm. is is oen a very subtle consideration, requiring help from a numerical
analyst or some other kind of magician.

we get a wrong answer. Instead we will use search via the following
procedure, which checks to see which of the endpoints has a negative
function value and which has a positive value, and calls the search pro-
cedure accordingly. If the function has the same sign on the two given
points, the half-interval method cannot be used, in which case the pro-
cedure signals an error.56
(define (half-interval-method f a b)
  (let ((a-value (f a))
          (b-value (f b)))
     (cond ((and (negative? a-value) (positive? b-value))
              (search f a b))
            ((and (negative? b-value) (positive? a-value))
              (search f b a))
              (error "Values are not of opposite sign" a b)))))

e following example uses the half-interval method to approximate π
as the root between 2 and 4 of sin x = 0:
(half-interval-method sin 2.0 4.0)

Here is another example, using the half-interval method to search for a
root of the equation x 3 − 2x − 3 = 0 between 1 and 2:
(half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))

  56 is can be accomplished using error, which takes as arguments a number of items

that are printed as error messages.

Finding fixed points of functions
A number x is called a fixed point of a function f if x satisfies the equa-
tion f (x) = x. For some functions f we can locate a fixed point by
beginning with an initial guess and applying f repeatedly,

                    f (x),   f (f (x)),    f (f (f (x))),   ...,

until the value does not change very much. Using this idea, we can de-
vise a procedure fixed-point that takes as inputs a function and an
initial guess and produces an approximation to a fixed point of the func-
tion. We apply the function repeatedly until we find two successive val-
ues whose difference is less than some prescribed tolerance:
(define tolerance 0.00001)
(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
     (< (abs (- v1 v2))
  (define (try guess)
     (let ((next (f guess)))
       (if (close-enough? guess next)
            (try next))))
  (try first-guess))

For example, we can use this method to approximate the fixed point of
the cosine function, starting with 1 as an initial approximation:57
(fixed-point cos 1.0)

Similarly, we can find a solution to the equation y = sin y + cos y:
  57 Try this during a boring lecture: Set your calculator to radians mode and then

repeatedly press the cos buon until you obtain the fixed point.

(fixed-point (lambda (y) (+ (sin y) (cos y)))

e fixed-point process is reminiscent of the process we used for finding
square roots in Section 1.1.7. Both are based on the idea of repeatedly
improving a guess until the result satisfies some criterion. In fact, we can
readily formulate the square-root computation as a fixed-point search.
Computing the square root of some number x requires finding a y such
that y 2 = x. Puing this equation into the equivalent form y = x/y,
we recognize that we are looking for a fixed point of the function58
y 7→ x/y, and we can therefore try to compute square roots as
(define (sqrt x)
  (fixed-point (lambda (y) (/ x y))

Unfortunately, this fixed-point search does not converge. Consider an
initial guess y1 . e next guess is y2 = x/y1 and the next guess is y3 =
x/y2 = x/(x/y1 ) = y1 . is results in an infinite loop in which the two
guesses y1 and y2 repeat over and over, oscillating about the answer.
     One way to control such oscillations is to prevent the guesses from
changing so much. Since the answer is always between our guess y and
x/y, we can make a new guess that is not as far from y as x/y by av-
eraging y with x/y, so that the next guess aer y is 21 (y + x/y) instead
of x/y. e process of making such a sequence of guesses is simply the
process of looking for a fixed point of y 7→ 12 (y + x/y):
(define (sqrt x)
  (fixed-point (lambda (y) (average y (/ x y)))

 58 7→ (pronounced “maps to”) is the mathematician’s way of writing lambda. y 7→ x/y

means (lambda (y) (/ x y)), that is, the function whose value at y is x/y.

(Note that y = 12 (y + x/y) is a simple transformation of the equation
y = x/y; to derive it, add y to both sides of the equation and divide by
    With this modification, the square-root procedure works. In fact, if
we unravel the definitions, we can see that the sequence of approxi-
mations to the square root generated here is precisely the same as the
one generated by our original square-root procedure of Section 1.1.7.
is approach of averaging successive approximations to a solution, a
technique that we call average damping, oen aids the convergence of
fixed-point searches.

      Exercise 1.35: Show that the golden ratio ϕ (Section 1.2.2)
      is a fixed point of the transformation x 7→ 1 + 1/x , and
      use this fact to compute ϕ by means of the fixed-point

      Exercise 1.36: Modify fixed-point so that it prints the
      sequence of approximations it generates, using the newline
      and display primitives shown in Exercise 1.22. en find
      a solution to x x = 1000 by finding a fixed point of x 7→
      log(1000)/ log(x). (Use Scheme’s primitive log procedure,
      which computes natural logarithms.) Compare the number
      of steps this takes with and without average damping. (Note
      that you cannot start fixed-point with a guess of 1, as this
      would cause division by log(1) = 0.)

      Exercise 1.37:

        a. An infinite continued fraction is an expression of the

                 f =                                 .
                       D1 +
                              D2 +
                                        D3 + . . .
As an example, one can show that the infinite con-
tinued fraction expansion with the Ni and the Di all
equal to 1 produces 1/ϕ, where ϕ is the golden ratio
(described in Section 1.2.2). One way to approximate
an infinite continued fraction is to truncate the expan-
sion aer a given number of terms. Such a truncation—
a so-called k-term finite continued fraction—has the form

                       D1 +
                              ..        Nk
Suppose that n and d are procedures of one argument
(the term index i) that return the Ni and Di of the
terms of the continued fraction. Define a procedure
cont-frac such that evaluating (cont-frac n d k)
computes the value of the k-term finite continued frac-
tion. Check your procedure by approximating 1/ϕ us-
(cont-frac (lambda (i) 1.0)
            (lambda (i) 1.0)

     for successive values of k. How large must you make
     k in order to get an approximation that is accurate to
     4 decimal places?
  b. If your cont-frac procedure generates a recursive pro-
     cess, write one that generates an iterative process. If
     it generates an iterative process, write one that gen-
     erates a recursive process.

Exercise 1.38: In 1737, the Swiss mathematician Leonhard
Euler published a memoir De Fractionibus Continuis, which
included a continued fraction expansion for e − 2, where
e is the base of the natural logarithms. In this fraction, the
Ni are all 1, and the Di are successively 1, 2, 1, 1, 4, 1, 1,
6, 1, 1, 8, . . .. Write a program that uses your cont-frac
procedure from Exercise 1.37 to approximate e, based on
Euler’s expansion.

Exercise 1.39: A continued fraction representation of the
tangent function was published in 1770 by the German math-
ematician J.H. Lambert:

                  tan x =                     ,
where x is in radians. Define a procedure (tan-cf x k) that
computes an approximation to the tangent function based
on Lambert’s formula. k specifies the number of terms to
compute, as in Exercise 1.37.

1.3.4 Procedures as Returned Values
e above examples demonstrate how the ability to pass procedures as
arguments significantly enhances the expressive power of our program-
ming language. We can achieve even more expressive power by creating
procedures whose returned values are themselves procedures.
     We can illustrate this idea by looking again at the fixed-point exam-
ple described at the end of Section 1.3.3. We formulated a new version
of the square-root procedure as a fixed-point search, starting with the
observation that x is a fixed-point of the function y 7→ x/y. en we
used average damping to make the approximations converge. Average
damping is a useful general technique in itself. Namely, given a function
f , we consider the function whose value at x is equal to the average of
x and f (x ).
     We can express the idea of average damping by means of the fol-
lowing procedure:
(define (average-damp f)
  (lambda (x) (average x (f x))))

average-damp   is a procedure that takes as its argument a procedure
f and returns as its value a procedure (produced by the lambda) that,
when applied to a number x, produces the average of x and (f x). For
example, applying average-damp to the square procedure produces a
procedure whose value at some number x is the average of x and x 2 .
Applying this resulting procedure to 10 returns the average of 10 and
100, or 55:59

  59 Observe that this is a combination whose operator is itself a combination. Exercise

1.4 already demonstrated the ability to form such combinations, but that was only a toy
example. Here we begin to see the real need for such combinations—when applying a
procedure that is obtained as the value returned by a higher-order procedure.

((average-damp square) 10)

Using average-damp, we can reformulate the square-root procedure as
(define (sqrt x)
  (fixed-point (average-damp (lambda (y) (/ x y)))

Notice how this formulation makes explicit the three ideas in the method:
fixed-point search, average damping, and the function y 7→ x/y. It is in-
structive to compare this formulation of the square-root method with
the original version given in Section 1.1.7. Bear in mind that these pro-
cedures express the same process, and notice how much clearer the idea
becomes when we express the process in terms of these abstractions. In
general, there are many ways to formulate a process as a procedure. Ex-
perienced programmers know how to choose procedural formulations
that are particularly perspicuous, and where useful elements of the pro-
cess are exposed as separate entities that can be reused in other appli-
cations. As a simple example of reuse, notice that the cube root of x is a
fixed point of the function y 7→ x/y 2 , so we can immediately generalize
our square-root procedure to one that extracts cube roots:60
(define (cube-root x)
  (fixed-point (average-damp (lambda (y) (/ x (square y))))

Newton’s method
When we first introduced the square-root procedure, in Section 1.1.7, we
mentioned that this was a special case of Newton’s method. If x 7→ д(x )
  60 See   Exercise 1.45 for a further generalization.

is a differentiable function, then a solution of the equation д(x) = 0 is a
fixed point of the function x 7→ f (x ), where
                                f (x ) = x −
and Dд(x) is the derivative of д evaluated at x. Newton’s method is the
use of the fixed-point method we saw above to approximate a solution
of the equation by finding a fixed point of the function f.61
     For many functions д and for sufficiently good initial guesses for x ,
Newton’s method converges very rapidly to a solution of д(x ) = 0.62
     In order to implement Newton’s method as a procedure, we must
first express the idea of derivative. Note that “derivative,” like average
damping, is something that transforms a function into another function.
For instance, the derivative of the function x 7→ x 3 is the function x 7→
3x 2. In general, if д is a function and dx is a small number, then the
derivative Dд of д is the function whose value at any number x is given
(in the limit of small dx) by
                                      д(x + dx) − д(x)
                           Dд(x) =                     .
us, we can express the idea of derivative (taking dx to be, say, 0.00001)
as the procedure
(define (deriv g)
  (lambda (x) (/ (- (g (+ x dx)) (g x)) dx)))

  61 Elementary  calculus books usually describe Newton’s method in terms of the se-
quence of approximations xn+1 = xn −д(xn )/Dд(xn ). Having language for talking about
processes and using the idea of fixed points simplifies the description of the method.
  62 Newton’s method does not always converge to an answer, but it can be shown

that in favorable cases each iteration doubles the number-of-digits accuracy of the ap-
proximation to the solution. In such cases, Newton’s method will converge much more
rapidly than the half-interval method.

along with the definition
(define dx 0.00001)

Like average-damp, deriv is a procedure that takes a procedure as ar-
gument and returns a procedure as value. For example, to approximate
the derivative of x 7→ x 3 at 5 (whose exact value is 75) we can evaluate
(define (cube x) (* x x x))
((deriv cube) 5)

With the aid of deriv, we can express Newton’s method as a fixed-point
(define (newton-transform g)
  (lambda (x) (- x (/ (g x) ((deriv g) x)))))
(define (newtons-method g guess)
  (fixed-point (newton-transform g) guess))

e newton-transform procedure expresses the formula at the begin-
ning of this section, and newtons-method is readily defined in terms of
this. It takes as arguments a procedure that computes the function for
which we want to find a zero, together with an initial guess. For in-
stance, to find the square root of x, we can use Newton’s method to find
a zero of the function y 7→ y 2 − x starting with an initial guess of 1.63
    is provides yet another form of the square-root procedure:
(define (sqrt x)
   (lambda (y) (- (square y) x)) 1.0))

  63 Forfinding square roots, Newton’s method converges rapidly to the correct solu-
tion from any starting point.

Abstractions and first-class procedures
We’ve seen two ways to express the square-root computation as an in-
stance of a more general method, once as a fixed-point search and once
using Newton’s method. Since Newton’s method was itself expressed
as a fixed-point process, we actually saw two ways to compute square
roots as fixed points. Each method begins with a function and finds a
fixed point of some transformation of the function. We can express this
general idea itself as a procedure:
(define (fixed-point-of-transform g transform guess)
  (fixed-point (transform g) guess))

is very general procedure takes as its arguments a procedure g that
computes some function, a procedure that transforms g, and an initial
guess. e returned result is a fixed point of the transformed function.
    Using this abstraction, we can recast the first square-root computa-
tion from this section (where we look for a fixed point of the average-
damped version of y 7→ x/y) as an instance of this general method:
(define (sqrt x)
   (lambda (y) (/ x y)) average-damp 1.0))

Similarly, we can express the second square-root computation from this
section (an instance of Newton’s method that finds a fixed point of the
Newton transform of y 7→ y 2 − x) as
(define (sqrt x)
   (lambda (y) (- (square y) x)) newton-transform 1.0))

We began Section 1.3 with the observation that compound procedures
are a crucial abstraction mechanism, because they permit us to express
general methods of computing as explicit elements in our programming

language. Now we’ve seen how higher-order procedures permit us to
manipulate these general methods to create further abstractions.
    As programmers, we should be alert to opportunities to identify the
underlying abstractions in our programs and to build upon them and
generalize them to create more powerful abstractions. is is not to
say that one should always write programs in the most abstract way
possible; expert programmers know how to choose the level of abstrac-
tion appropriate to their task. But it is important to be able to think in
terms of these abstractions, so that we can be ready to apply them in
new contexts. e significance of higher-order procedures is that they
enable us to represent these abstractions explicitly as elements in our
programming language, so that they can be handled just like other com-
putational elements.
    In general, programming languages impose restrictions on the ways
in which computational elements can be manipulated. Elements with
the fewest restrictions are said to have first-class status. Some of the
“rights and privileges” of first-class elements are:64

    • ey may be named by variables.

    • ey may be passed as arguments to procedures.

    • ey may be returned as the results of procedures.

    • ey may be included in data structures.65

Lisp, unlike other common programming languages, awards procedures
full first-class status. is poses challenges for efficient implementation,
  64 e  notion of first-class status of programming-language elements is due to the
British computer scientist Christopher Strachey (1916-1975).
  65 We’ll see examples of this aer we introduce data structures in Chapter 2.

but the resulting gain in expressive power is enormous.66

       Exercise 1.40: Define a procedure cubic that can be used
       together with the newtons-method procedure in expressions
       of the form
       (newtons-method (cubic a b c) 1)

       to approximate zeros of the cubic x 3 + ax 2 + bx + c.

       Exercise 1.41: Define a procedure double that takes a pro-
       cedure of one argument as argument and returns a proce-
       dure that applies the original procedure twice. For exam-
       ple, if inc is a procedure that adds 1 to its argument, then
       (double inc) should be a procedure that adds 2. What
       value is returned by
       (((double (double double)) inc) 5)

       Exercise 1.42: Let f and д be two one-argument functions.
       e composition f aer д is defined to be the function x 7→
       f (д(x)). Define a procedure compose that implements com-
       position. For example, if inc is a procedure that adds 1 to
       its argument,
       ((compose square inc) 6)

   66 e major implementation cost of first-class procedures is that allowing procedures

to be returned as values requires reserving storage for a procedure’s free variables even
while the procedure is not executing. In the Scheme implementation we will study in
Section 4.1, these variables are stored in the procedure’s environment.

Exercise 1.43: If f is a numerical function and n is a posi-
tive integer, then we can form the n th repeated application
of f , which is defined to be the function whose value at
x is f (f (. . . (f (x )) . . . )). For example, if f is the function
x 7→ x + 1, then the n th repeated application of f is the
function x 7→ x +n. If f is the operation of squaring a num-
ber, then the n th repeated application of f is the function
that raises its argument to the 2n -th power. Write a proce-
dure that takes as inputs a procedure that computes f and a
positive integer n and returns the procedure that computes
the n th repeated application of f . Your procedure should be
able to be used as follows:
((repeated square 2) 5)

Hint: You may find it convenient to use compose from Ex-
ercise 1.42.

Exercise 1.44: e idea of smoothing a function is an im-
portant concept in signal processing. If f is a function and
dx is some small number, then the smoothed version of f is
the function whose value at a point x is the average of f (x −
dx), f (x), and f (x +dx). Write a procedure smooth that takes
as input a procedure that computes f and returns a proce-
dure that computes the smoothed f . It is sometimes valu-
able to repeatedly smooth a function (that is, smooth the
smoothed function, and so on) to obtain the n-fold smoothed
function. Show how to generate the n-fold smoothed func-
tion of any given function using smooth and repeated from
Exercise 1.43.

Exercise 1.45: We saw in Section 1.3.3 that aempting to
compute square roots by naively finding a fixed point of
y 7→ x/y does not converge, and that this can be fixed by
average damping. e same method works for finding cube
roots as fixed points of the average-damped y 7→ x/y 2 . Un-
fortunately, the process does not work for fourth roots—a
single average damp is not enough to make a fixed-point
search for y 7→ x/y 3 converge. On the other hand, if we
average damp twice (i.e., use the average damp of the av-
erage damp of y 7→ x/y 3 ) the fixed-point search does con-
verge. Do some experiments to determine how many av-
erage damps are required to compute n th roots as a fixed-
point search based upon repeated average damping of y 7→
x/yn −1 . Use this to implement a simple procedure for com-
puting n th roots using fixed-point, average-damp, and the
repeated procedure of Exercise 1.43. Assume that any arith-
metic operations you need are available as primitives.

Exercise 1.46: Several of the numerical methods described
in this chapter are instances of an extremely general com-
putational strategy known as iterative improvement. Itera-
tive improvement says that, to compute something, we start
with an initial guess for the answer, test if the guess is good
enough, and otherwise improve the guess and continue the
process using the improved guess as the new guess. Write
a procedure iterative-improve that takes two procedures
as arguments: a method for telling whether a guess is good
enough and a method for improving a guess. iterative-
improve should return as its value a procedure that takes a
guess as argument and keeps improving the guess until it is

good enough. Rewrite the sqrt procedure of Section 1.1.7
and the fixed-point procedure of Section 1.3.3 in terms of

Building Abstractions with Data

     We now come to the decisive step of mathematical abstrac-
     tion: we forget about what the symbols stand for. . . .[e
     mathematician] need not be idle; there are many operations
     which he may carry out with these symbols, without ever
     having to look at the things they stand for.
     —Hermann Weyl, e Mathematical Way of inking

W        Chapter 1 on computational processes and
      on the role of procedures in program design. We saw how to
use primitive data (numbers) and primitive operations (arithmetic op-
erations), how to combine procedures to form compound procedures
through composition, conditionals, and the use of parameters, and how
to abstract procedures by using define. We saw that a procedure can
be regarded as a paern for the local evolution of a process, and we
classified, reasoned about, and performed simple algorithmic analyses
of some common paerns for processes as embodied in procedures. We

also saw that higher-order procedures enhance the power of our lan-
guage by enabling us to manipulate, and thereby to reason in terms of,
general methods of computation. is is much of the essence of pro-
    In this chapter we are going to look at more complex data. All the
procedures in chapter 1 operate on simple numerical data, and simple
data are not sufficient for many of the problems we wish to address
using computation. Programs are typically designed to model complex
phenomena, and more oen than not one must construct computational
objects that have several parts in order to model real-world phenom-
ena that have several aspects. us, whereas our focus in chapter 1 was
on building abstractions by combining procedures to form compound
procedures, we turn in this chapter to another key aspect of any pro-
gramming language: the means it provides for building abstractions by
combining data objects to form compound data.
    Why do we want compound data in a programming language? For
the same reasons that we want compound procedures: to elevate the
conceptual level at which we can design our programs, to increase the
modularity of our designs, and to enhance the expressive power of our
language. Just as the ability to define procedures enables us to deal with
processes at a higher conceptual level than that of the primitive oper-
ations of the language, the ability to construct compound data objects
enables us to deal with data at a higher conceptual level than that of the
primitive data objects of the language.
    Consider the task of designing a system to perform arithmetic with
rational numbers. We could imagine an operation add-rat that takes
two rational numbers and produces their sum. In terms of simple data,
a rational number can be thought of as two integers: a numerator and
a denominator. us, we could design a program in which each ratio-
nal number would be represented by two integers (a numerator and a

denominator) and where add-rat would be implemented by two proce-
dures (one producing the numerator of the sum and one producing the
denominator). But this would be awkward, because we would then need
to explicitly keep track of which numerators corresponded to which de-
nominators. In a system intended to perform many operations on many
rational numbers, such bookkeeping details would cluer the programs
substantially, to say nothing of what they would do to our minds. It
would be much beer if we could “glue together” a numerator and de-
nominator to form a pair—a compound data object —that our programs
could manipulate in a way that would be consistent with regarding a
rational number as a single conceptual unit.
    e use of compound data also enables us to increase the modular-
ity of our programs. If we can manipulate rational numbers directly as
objects in their own right, then we can separate the part of our program
that deals with rational numbers per se from the details of how rational
numbers may be represented as pairs of integers. e general technique
of isolating the parts of a program that deal with how data objects are
represented from the parts of a program that deal with how data objects
are used is a powerful design methodology called data abstraction. We
will see how data abstraction makes programs much easier to design,
maintain, and modify.
    e use of compound data leads to a real increase in the expressive
power of our programming language. Consider the idea of forming a
“linear combination” ax + by. We might like to write a procedure that
would accept a, b, x, and y as arguments and return the value of ax +by.
is presents no difficulty if the arguments are to be numbers, because
we can readily define the procedure
(define (linear-combination a b x y)
  (+ (* a x) (* b y)))

But suppose we are not concerned only with numbers. Suppose we
would like to express, in procedural terms, the idea that one can form
linear combinations whenever addition and multiplication are defined—
for rational numbers, complex numbers, polynomials, or whatever. We
could express this as a procedure of the form
(define (linear-combination a b x y)
  (add (mul a x) (mul b y)))

where add and mul are not the primitive procedures + and * but rather
more complex things that will perform the appropriate operations for
whatever kinds of data we pass in as the arguments a, b, x, and y. e
key point is that the only thing linear-combination should need to
know about a, b, x, and y is that the procedures add and mul will per-
form the appropriate manipulations. From the perspective of the pro-
cedure linear-combination, it is irrelevant what a, b, x, and y are and
even more irrelevant how they might happen to be represented in terms
of more primitive data. is same example shows why it is important
that our programming language provide the ability to manipulate com-
pound objects directly: Without this, there is no way for a procedure
such as linear-combination to pass its arguments along to add and
mul without having to know their detailed structure.1
   1 e ability to directly manipulate procedures provides an analogous increase in the

expressive power of a programming language. For example, in Section 1.3.1 we intro-
duced the sum procedure, which takes a procedure term as an argument and computes
the sum of the values of term over some specified interval. In order to define sum, it
is crucial that we be able to speak of a procedure such as term as an entity in its own
right, without regard for how term might be expressed with more primitive operations.
Indeed, if we did not have the notion of “a procedure,” it is doubtful that we would ever
even think of the possibility of defining an operation such as sum. Moreover, insofar as
performing the summation is concerned, the details of how term may be constructed
from more primitive operations are irrelevant.

    We begin this chapter by implementing the rational-number arith-
metic system mentioned above. is will form the background for our
discussion of compound data and data abstraction. As with compound
procedures, the main issue to be addressed is that of abstraction as a
technique for coping with complexity, and we will see how data ab-
straction enables us to erect suitable abstraction barriers between differ-
ent parts of a program.
    We will see that the key to forming compound data is that a pro-
gramming language should provide some kind of “glue” so that data
objects can be combined to form more complex data objects. ere are
many possible kinds of glue. Indeed, we will discover how to form com-
pound data using no special “data” operations at all, only procedures.
is will further blur the distinction between “procedure” and “data,”
which was already becoming tenuous toward the end of chapter 1. We
will also explore some conventional techniques for representing sequences
and trees. One key idea in dealing with compound data is the notion of
closure—that the glue we use for combining data objects should allow
us to combine not only primitive data objects, but compound data ob-
jects as well. Another key idea is that compound data objects can serve
as conventional interfaces for combining program modules in mix-and-
match ways. We illustrate some of these ideas by presenting a simple
graphics language that exploits closure.
    We will then augment the representational power of our language
by introducing symbolic expressions—data whose elementary parts can
be arbitrary symbols rather than only numbers. We explore various al-
ternatives for representing sets of objects. We will find that, just as a
given numerical function can be computed by many different computa-
tional processes, there are many ways in which a given data structure
can be represented in terms of simpler objects, and the choice of rep-
resentation can have significant impact on the time and space require-

ments of processes that manipulate the data. We will investigate these
ideas in the context of symbolic differentiation, the representation of
sets, and the encoding of information.
    Next we will take up the problem of working with data that may be
represented differently by different parts of a program. is leads to the
need to implement generic operations, which must handle many different
types of data. Maintaining modularity in the presence of generic oper-
ations requires more powerful abstraction barriers than can be erected
with simple data abstraction alone. In particular, we introduce data-
directed programming as a technique that allows individual data repre-
sentations to be designed in isolation and then combined additively (i.e.,
without modification). To illustrate the power of this approach to sys-
tem design, we close the chapter by applying what we have learned to
the implementation of a package for performing symbolic arithmetic on
polynomials, in which the coefficients of the polynomials can be inte-
gers, rational numbers, complex numbers, and even other polynomials.

2.1 Introduction to Data Abstraction
In Section 1.1.8, we noted that a procedure used as an element in creat-
ing a more complex procedure could be regarded not only as a collection
of particular operations but also as a procedural abstraction. at is, the
details of how the procedure was implemented could be suppressed,
and the particular procedure itself could be replaced by any other pro-
cedure with the same overall behavior. In other words, we could make
an abstraction that would separate the way the procedure would be used
from the details of how the procedure would be implemented in terms
of more primitive procedures. e analogous notion for compound data
is called data abstraction. Data abstraction is a methodology that enables

us to isolate how a compound data object is used from the details of how
it is constructed from more primitive data objects.
     e basic idea of data abstraction is to structure the programs that
are to use compound data objects so that they operate on “abstract data.”
at is, our programs should use data in such a way as to make no as-
sumptions about the data that are not strictly necessary for performing
the task at hand. At the same time, a “concrete” data representation is
defined independent of the programs that use the data. e interface be-
tween these two parts of our system will be a set of procedures, called se-
lectors and constructors, that implement the abstract data in terms of the
concrete representation. To illustrate this technique, we will consider
how to design a set of procedures for manipulating rational numbers.

2.1.1 Example: Arithmetic Operations
      for Rational Numbers
Suppose we want to do arithmetic with rational numbers. We want to
be able to add, subtract, multiply, and divide them and to test whether
two rational numbers are equal.
    Let us begin by assuming that we already have a way of construct-
ing a rational number from a numerator and a denominator. We also
assume that, given a rational number, we have a way of extracting (or
selecting) its numerator and its denominator. Let us further assume that
the constructor and selectors are available as procedures:

    • (make-rat ⟨n⟩ ⟨d ⟩) returns the rational number whose numera-
      tor is the integer ⟨n⟩ and whose denominator is the integer ⟨d ⟩.

    • (numer ⟨x ⟩) returns the numerator of the rational number ⟨x⟩.

    • (denom ⟨x ⟩) returns the denominator of the rational number ⟨x ⟩.

We are using here a powerful strategy of synthesis: wishful thinking.
We haven’t yet said how a rational number is represented, or how the
procedures numer, denom, and make-rat should be implemented. Even
so, if we did have these three procedures, we could then add, subtract,
multiply, divide, and test equality by using the following relations:
             n1 n2            n 1d 2 + n 2d 1
                +         =                   ,
             d1 d2                d 1d 2
             n1 n2            n 1d 2 − n 2d 1
                −         =                    ,
             d1 d2                 d 1d 2
              n1 n2           n 1n 2
                  ·       =          ,
              d1 d2           d 1d 2
               n 1 /d 1       n 1d 2
                          =          ,
               n 2 /d 2       d 1n 2
                    n1        n2
                          =            if and only if n 1d 2 = n 2d 1 .
                    d1        d2
We can express these rules as procedures:
(define (add-rat x y)
  (make-rat (+ (* (numer x) (denom y))
                 (* (numer y) (denom x)))
             (* (denom x) (denom y))))
(define (sub-rat x y)
  (make-rat (- (* (numer x) (denom y))
                 (* (numer y) (denom x)))
             (* (denom x) (denom y))))
(define (mul-rat x y)
  (make-rat (* (numer x) (numer y))
             (* (denom x) (denom y))))
(define (div-rat x y)
  (make-rat (* (numer x) (denom y))
             (* (denom x) (numer y))))

(define (equal-rat? x y)
    (= (* (numer x) (denom y))
       (* (numer y) (denom x))))

    Now we have the operations on rational numbers defined in terms
of the selector and constructor procedures numer, denom, and make-rat.
But we haven’t yet defined these. What we need is some way to glue
together a numerator and a denominator to form a rational number.

To enable us to implement the concrete level of our data abstraction,
our language provides a compound structure called a pair, which can
be constructed with the primitive procedure cons. is procedure takes
two arguments and returns a compound data object that contains the
two arguments as parts. Given a pair, we can extract the parts using the
primitive procedures car and cdr.2 us, we can use cons, car, and cdr
as follows:
(define x (cons 1 2))
(car x)
(cdr x)

Notice that a pair is a data object that can be given a name and manip-
ulated, just like a primitive data object. Moreover, cons can be used to
form pairs whose elements are pairs, and so on:
    2 e name cons stands for “construct.” e names car and cdr derive from the orig-

inal implementation of Lisp on the  . at machine had an addressing scheme
that allowed one to reference the “address” and “decrement” parts of a memory location.
car stands for “Contents of Address part of Register” and cdr (pronounced “could-er”)
stands for “Contents of Decrement part of Register.”

(define x (cons 1 2))
(define y (cons 3 4))
(define z (cons x y))
(car (car z))
(car (cdr z))

In Section 2.2 we will see how this ability to combine pairs means that
pairs can be used as general-purpose building blocks to create all sorts
of complex data structures. e single compound-data primitive pair,
implemented by the procedures cons, car, and cdr, is the only glue we
need. Data objects constructed from pairs are called list-structured data.

Representing rational numbers
Pairs offer a natural way to complete the rational-number system. Sim-
ply represent a rational number as a pair of two integers: a numerator
and a denominator. en make-rat, numer, and denom are readily im-
plemented as follows:3
    3 Another   way to define the selectors and constructor is
(define make-rat cons)
(define numer car)
(define denom cdr)

  e first definition associates the name make-rat with the value of the expression
cons,  which is the primitive procedure that constructs pairs. us make-rat and cons
are names for the same primitive constructor.
  Defining selectors and constructors in this way is efficient: Instead of make-rat call-
ing cons, make-rat is cons, so there is only one procedure called, not two, when make-
rat is called. On the other hand, doing this defeats debugging aids that trace procedure
calls or put breakpoints on procedure calls: You may want to watch make-rat being
called, but you certainly don’t want to watch every call to cons.
  We have chosen not to use this style of definition in this book.

(define (make-rat n d) (cons n d))
(define (numer x) (car x))
(define (denom x) (cdr x))

Also, in order to display the results of our computations, we can print
rational numbers by printing the numerator, a slash, and the denomi-
(define (print-rat x)
  (display (numer x))
  (display "/")
  (display (denom x)))

Now we can try our rational-number procedures:
(define one-half (make-rat 1 2))
(print-rat one-half)
(define one-third (make-rat 1 3))
(print-rat (add-rat one-half one-third))
(print-rat (mul-rat one-half one-third))
(print-rat (add-rat one-third one-third))

As the final example shows, our rational-number implementation does
not reduce rational numbers to lowest terms. We can remedy this by
changing make-rat. If we have a gcd procedure like the one in Section
1.2.5 that produces the greatest common divisor of two integers, we can
   4 display  is the Scheme primitive for printing data. e Scheme primitive newline
starts a new line for printing. Neither of these procedures returns a useful value, so
in the uses of print-rat below, we show only what print-rat prints, not what the
interpreter prints as the value returned by print-rat.

use gcd to reduce the numerator and the denominator to lowest terms
before constructing the pair:
(define (make-rat n d)
  (let ((g (gcd n d)))
      (cons (/ n g) (/ d g))))

Now we have
(print-rat (add-rat one-third one-third))

as desired. is modification was accomplished by changing the con-
structor make-rat without changing any of the procedures (such as
add-rat and mul-rat) that implement the actual operations.

       Exercise 2.1: Define a beer version of make-rat that han-
       dles both positive and negative arguments. make-rat should
       normalize the sign so that if the rational number is positive,
       both the numerator and denominator are positive, and if
       the rational number is negative, only the numerator is neg-

2.1.2 Abstraction Barriers
Before continuing with more examples of compound data and data ab-
straction, let us consider some of the issues raised by the rational-number
example. We defined the rational-number operations in terms of a con-
structor make-rat and selectors numer and denom. In general, the under-
lying idea of data abstraction is to identify for each type of data object
a basic set of operations in terms of which all manipulations of data
objects of that type will be expressed, and then to use only those oper-
ations in manipulating the data.

                 Programs that use rational numbers

                   Rational numbers in problem domain

                        add-rat    sub-rat       ...

             Rational numbers as numerators and denominators

                        make-rat   numer     denom

                        Rational numbers as pairs

                            cons   car     cdr

                      However pairs are implemented

      Figure 2.1: Data-abstraction barriers in the rational-
      number package.

    We can envision the structure of the rational-number system as
shown in Figure 2.1. e horizontal lines represent abstraction barriers
that isolate different “levels” of the system. At each level, the barrier
separates the programs (above) that use the data abstraction from the
programs (below) that implement the data abstraction. Programs that
use rational numbers manipulate them solely in terms of the proce-
dures supplied “for public use” by the rational-number package: add-
rat, sub-rat, mul-rat, div-rat, and equal-rat?. ese, in turn, are
implemented solely in terms of the constructor and selectors make-rat,
numer, and denom, which themselves are implemented in terms of pairs.
e details of how pairs are implemented are irrelevant to the rest of
the rational-number package so long as pairs can be manipulated by
the use of cons, car, and cdr. In effect, procedures at each level are the

interfaces that define the abstraction barriers and connect the different
    is simple idea has many advantages. One advantage is that it
makes programs much easier to maintain and to modify. Any complex
data structure can be represented in a variety of ways with the prim-
itive data structures provided by a programming language. Of course,
the choice of representation influences the programs that operate on it;
thus, if the representation were to be changed at some later time, all
such programs might have to be modified accordingly. is task could
be time-consuming and expensive in the case of large programs unless
the dependence on the representation were to be confined by design to
a very few program modules.
    For example, an alternate way to address the problem of reducing
rational numbers to lowest terms is to perform the reduction whenever
we access the parts of a rational number, rather than when we construct
it. is leads to different constructor and selector procedures:
(define (make-rat n d) (cons n d))
(define (numer x)
  (let ((g (gcd (car x) (cdr x))))
    (/ (car x) g)))
(define (denom x)
  (let ((g (gcd (car x) (cdr x))))
    (/ (cdr x) g)))

e difference between this implementation and the previous one lies
in when we compute the gcd. If in our typical use of rational numbers
we access the numerators and denominators of the same rational num-
bers many times, it would be preferable to compute the gcd when the
rational numbers are constructed. If not, we may be beer off waiting
until access time to compute the gcd. In any case, when we change from

one representation to the other, the procedures add-rat, sub-rat, and
so on do not have to be modified at all.
    Constraining the dependence on the representation to a few in-
terface procedures helps us design programs as well as modify them,
because it allows us to maintain the flexibility to consider alternate
implementations. To continue with our simple example, suppose we
are designing a rational-number package and we can’t decide initially
whether to perform the gcd at construction time or at selection time.
e data-abstraction methodology gives us a way to defer that decision
without losing the ability to make progress on the rest of the system.

      Exercise 2.2: Consider the problem of representing line
      segments in a plane. Each segment is represented as a pair
      of points: a starting point and an ending point. Define a
      constructor make-segment and selectors start-segment and
      end-segment that define the representation of segments in
      terms of points. Furthermore, a point can be represented
      as a pair of numbers: the x coordinate and the y coordi-
      nate. Accordingly, specify a constructor make-point and
      selectors x-point and y-point that define this representa-
      tion. Finally, using your selectors and constructors, define a
      procedure midpoint-segment that takes a line segment as
      argument and returns its midpoint (the point whose coor-
      dinates are the average of the coordinates of the endpoints).
      To try your procedures, you’ll need a way to print points:
      (define (print-point p)
        (display "(")
        (display (x-point p))
        (display ",")

        (display (y-point p))
        (display ")"))

      Exercise 2.3: Implement a representation for rectangles in
      a plane. (Hint: You may want to make use of Exercise 2.2.) In
      terms of your constructors and selectors, create procedures
      that compute the perimeter and the area of a given rectan-
      gle. Now implement a different representation for rectan-
      gles. Can you design your system with suitable abstraction
      barriers, so that the same perimeter and area procedures
      will work using either representation?

2.1.3 What Is Meant by Data?
We began the rational-number implementation in Section 2.1.1 by im-
plementing the rational-number operations add-rat, sub-rat, and so
on in terms of three unspecified procedures: make-rat, numer, and denom.
At that point, we could think of the operations as being defined in terms
of data objects—numerators, denominators, and rational numbers—whose
behavior was specified by the laer three procedures.
    But exactly what is meant by data? It is not enough to say “whatever
is implemented by the given selectors and constructors.” Clearly, not
every arbitrary set of three procedures can serve as an appropriate basis
for the rational-number implementation. We need to guarantee that, if
we construct a rational number x from a pair of integers n and d, then
extracting the numer and the denom of x and dividing them should yield
the same result as dividing n by d. In other words, make-rat, numer,
and denom must satisfy the condition that, for any integer n and any

non-zero integer d, if x is (make-rat n d), then
                                  (numer x)          n
                                                 =       .
                                  (denom x)          d

In fact, this is the only condition make-rat, numer, and denom must fulfill
in order to form a suitable basis for a rational-number representation.
In general, we can think of data as defined by some collection of se-
lectors and constructors, together with specified conditions that these
procedures must fulfill in order to be a valid representation.5
    is point of view can serve to define not only “high-level” data ob-
jects, such as rational numbers, but lower-level objects as well. Consider
the notion of a pair, which we used in order to define our rational num-
bers. We never actually said what a pair was, only that the language
supplied procedures cons, car, and cdr for operating on pairs. But the
only thing we need to know about these three operations is that if we
glue two objects together using cons we can retrieve the objects using
car and cdr. at is, the operations satisfy the condition that, for any
objects x and y, if z is (cons x y) then (car z) is x and (cdr z) is y.
   5 Surprisingly, this idea is very difficult to formulate rigorously. ere are two ap-
proaches to giving such a formulation. One, pioneered by C. A. R. Hoare (1972), is
known as the method of abstract models. It formalizes the “procedures plus conditions”
specification as outlined in the rational-number example above. Note that the condi-
tion on the rational-number representation was stated in terms of facts about integers
(equality and division). In general, abstract models define new kinds of data objects
in terms of previously defined types of data objects. Assertions about data objects can
therefore be checked by reducing them to assertions about previously defined data ob-
jects. Another approach, introduced by Zilles at , by Goguen, atcher, Wagner, and
Wright at  (see atcher et al. 1978), and by Guag at Toronto (see Guag 1977), is
called algebraic specification. It regards the “procedures” as elements of an abstract alge-
braic system whose behavior is specified by axioms that correspond to our “conditions,”
and uses the techniques of abstract algebra to check assertions about data objects. Both
methods are surveyed in the paper by Liskov and Zilles (1975).

Indeed, we mentioned that these three procedures are included as prim-
itives in our language. However, any triple of procedures that satisfies
the above condition can be used as the basis for implementing pairs.
is point is illustrated strikingly by the fact that we could implement
cons, car, and cdr without using any data structures at all but only
using procedures. Here are the definitions:
(define (cons x y)
  (define (dispatch m)
    (cond ((= m 0) x)
           ((= m 1) y)
           (else (error "Argument not 0 or 1: CONS" m))))
(define (car z) (z 0))
(define (cdr z) (z 1))

is use of procedures corresponds to nothing like our intuitive notion
of what data should be. Nevertheless, all we need to do to show that
this is a valid way to represent pairs is to verify that these procedures
satisfy the condition given above.
    e subtle point to notice is that the value returned by (cons x y) is
a procedure—namely the internally defined procedure dispatch, which
takes one argument and returns either x or y depending on whether the
argument is 0 or 1. Correspondingly, (car z) is defined to apply z to 0.
Hence, if z is the procedure formed by (cons x y), then z applied to 0
will yield x. us, we have shown that (car (cons x y)) yields x, as
desired. Similarly, (cdr (cons x y)) applies the procedure returned by
(cons x y) to 1, which returns y. erefore, this procedural implemen-
tation of pairs is a valid implementation, and if we access pairs using
only cons, car, and cdr we cannot distinguish this implementation from
one that uses “real” data structures.
    e point of exhibiting the procedural representation of pairs is not

that our language works this way (Scheme, and Lisp systems in general,
implement pairs directly, for efficiency reasons) but that it could work
this way. e procedural representation, although obscure, is a perfectly
adequate way to represent pairs, since it fulfills the only conditions that
pairs need to fulfill. is example also demonstrates that the ability to
manipulate procedures as objects automatically provides the ability to
represent compound data. is may seem a curiosity now, but procedu-
ral representations of data will play a central role in our programming
repertoire. is style of programming is oen called message passing,
and we will be using it as a basic tool in Chapter 3 when we address the
issues of modeling and simulation.

      Exercise 2.4: Here is an alternative procedural representa-
      tion of pairs. For this representation, verify that (car (cons
      x y)) yields x for any objects x and y.

      (define (cons x y)
        (lambda (m) (m x y)))
      (define (car z)
        (z (lambda (p q) p)))

      What is the corresponding definition of cdr? (Hint: To ver-
      ify that this works, make use of the substitution model of
      Section 1.1.5.)

      Exercise 2.5: Show that we can represent pairs of nonneg-
      ative integers using only numbers and arithmetic opera-
      tions if we represent the pair a and b as the integer that is
      the product 2a 3b . Give the corresponding definitions of the
      procedures cons, car, and cdr.

      Exercise 2.6: In case representing pairs as procedures wasn’t
      mind-boggling enough, consider that, in a language that
      can manipulate procedures, we can get by without numbers
      (at least insofar as nonnegative integers are concerned) by
      implementing 0 and the operation of adding 1 as
      (define zero (lambda (f) (lambda (x) x)))
      (define (add-1 n)
         (lambda (f) (lambda (x) (f ((n f) x)))))

      is representation is known as Church numerals, aer its
      inventor, Alonzo Church, the logician who invented the λ-
      Define one and two directly (not in terms of zero and add-
      1). (Hint: Use substitution to evaluate (add-1 zero)). Give
      a direct definition of the addition procedure + (not in terms
      of repeated application of add-1).

2.1.4 Extended Exercise: Interval Arithmetic
Alyssa P. Hacker is designing a system to help people solve engineer-
ing problems. One feature she wants to provide in her system is the
ability to manipulate inexact quantities (such as measured parameters
of physical devices) with known precision, so that when computations
are done with such approximate quantities the results will be numbers
of known precision.
    Electrical engineers will be using Alyssa’s system to compute elec-
trical quantities. It is sometimes necessary for them to compute the
value of a parallel equivalent resistance R p of two resistors R 1 , R 2 using
the formula
                            Rp =                .
                                  1/R 1 + 1/R 2

Resistance values are usually known only up to some tolerance guaran-
teed by the manufacturer of the resistor. For example, if you buy a resis-
tor labeled “6.8 ohms with 10% tolerance” you can only be sure that the
resistor has a resistance between 6.8 − 0.68 = 6.12 and 6.8 + 0.68 = 7.48
ohms. us, if you have a 6.8-ohm 10% resistor in parallel with a 4.7-ohm
5% resistor, the resistance of the combination can range from about 2.58
ohms (if the two resistors are at the lower bounds) to about 2.97 ohms
(if the two resistors are at the upper bounds).
     Alyssa’s idea is to implement “interval arithmetic” as a set of arith-
metic operations for combining “intervals” (objects that represent the
range of possible values of an inexact quantity). e result of adding,
subtracting, multiplying, or dividing two intervals is itself an interval,
representing the range of the result.
     Alyssa postulates the existence of an abstract object called an “in-
terval” that has two endpoints: a lower bound and an upper bound. She
also presumes that, given the endpoints of an interval, she can con-
struct the interval using the data constructor make-interval. Alyssa
first writes a procedure for adding two intervals. She reasons that the
minimum value the sum could be is the sum of the two lower bounds
and the maximum value it could be is the sum of the two upper bounds:
(define (add-interval x y)
  (make-interval (+ (lower-bound x) (lower-bound y))
                   (+ (upper-bound x) (upper-bound y))))

Alyssa also works out the product of two intervals by finding the min-
imum and the maximum of the products of the bounds and using them
as the bounds of the resulting interval. (min and max are primitives that
find the minimum or maximum of any number of arguments.)
(define (mul-interval x y)
  (let ((p1 (* (lower-bound x) (lower-bound y)))

          (p2 (* (lower-bound x) (upper-bound y)))
          (p3 (* (upper-bound x) (lower-bound y)))
          (p4 (* (upper-bound x) (upper-bound y))))
       (make-interval (min p1 p2 p3 p4)
                      (max p1 p2 p3 p4))))

To divide two intervals, Alyssa multiplies the first by the reciprocal of
the second. Note that the bounds of the reciprocal interval are the re-
ciprocal of the upper bound and the reciprocal of the lower bound, in
that order.
(define (div-interval x y)
   (make-interval (/ 1.0 (upper-bound y))
                     (/ 1.0 (lower-bound y)))))

        Exercise 2.7: Alyssa’s program is incomplete because she
        has not specified the implementation of the interval ab-
        straction. Here is a definition of the interval constructor:
        (define (make-interval a b) (cons a b))

        Define selectors upper-bound and lower-bound to complete
        the implementation.

        Exercise 2.8: Using reasoning analogous to Alyssa’s, de-
        scribe how the difference of two intervals may be com-
        puted. Define a corresponding subtraction procedure, called

        Exercise 2.9: e width of an interval is half of the differ-
        ence between its upper and lower bounds. e width is a

measure of the uncertainty of the number specified by the
interval. For some arithmetic operations the width of the
result of combining two intervals is a function only of the
widths of the argument intervals, whereas for others the
width of the combination is not a function of the widths of
the argument intervals. Show that the width of the sum (or
difference) of two intervals is a function only of the widths
of the intervals being added (or subtracted). Give examples
to show that this is not true for multiplication or division.

Exercise 2.10: Ben Bitdiddle, an expert systems program-
mer, looks over Alyssa’s shoulder and comments that it is
not clear what it means to divide by an interval that spans
zero. Modify Alyssa’s code to check for this condition and
to signal an error if it occurs.

Exercise 2.11: In passing, Ben also cryptically comments:
“By testing the signs of the endpoints of the intervals, it is
possible to break mul-interval into nine cases, only one
of which requires more than two multiplications.” Rewrite
this procedure using Ben’s suggestion.
Aer debugging her program, Alyssa shows it to a poten-
tial user, who complains that her program solves the wrong
problem. He wants a program that can deal with numbers
represented as a center value and an additive tolerance; for
example, he wants to work with intervals such as 3.5 ± 0.15
rather than [3.35, 3.65]. Alyssa returns to her desk and fixes
this problem by supplying an alternate constructor and al-
ternate selectors:

(define (make-center-width c w)
  (make-interval (- c w) (+ c w)))
(define (center i)
  (/ (+ (lower-bound i) (upper-bound i)) 2))
(define (width i)
  (/ (- (upper-bound i) (lower-bound i)) 2))

Unfortunately, most of Alyssa’s users are engineers. Real
engineering situations usually involve measurements with
only a small uncertainty, measured as the ratio of the width
of the interval to the midpoint of the interval. Engineers
usually specify percentage tolerances on the parameters of
devices, as in the resistor specifications given earlier.

Exercise 2.12: Define a constructor make-center-percent
that takes a center and a percentage tolerance and pro-
duces the desired interval. You must also define a selector
percent that produces the percentage tolerance for a given
interval. e center selector is the same as the one shown

Exercise 2.13: Show that under the assumption of small
percentage tolerances there is a simple formula for the ap-
proximate percentage tolerance of the product of two in-
tervals in terms of the tolerances of the factors. You may
simplify the problem by assuming that all numbers are pos-
Aer considerable work, Alyssa P. Hacker delivers her fin-
ished system. Several years later, aer she has forgoen all
about it, she gets a frenzied call from an irate user, Lem E.
Tweakit. It seems that Lem has noticed that the formula for

parallel resistors can be wrien in two algebraically equiv-
alent ways:
                            R1 R2
                           R1 + R2
                       1/R 1 + 1/R 2
He has wrien the following two programs, each of which
computes the parallel-resistors formula differently:
(define (par1 r1 r2)
  (div-interval (mul-interval r1 r2)
                 (add-interval r1 r2)))

(define (par2 r1 r2)
  (let ((one (make-interval 1 1)))
      one (add-interval (div-interval one r1)
                         (div-interval one r2)))))

Lem complains that Alyssa’s program gives different an-
swers for the two ways of computing. is is a serious com-

Exercise 2.14: Demonstrate that Lem is right. Investigate
the behavior of the system on a variety of arithmetic ex-
pressions. Make some intervals A and B, and use them in
computing the expressions A/A and A/B. You will get the
most insight by using intervals whose width is a small per-
centage of the center value. Examine the results of the com-
putation in center-percent form (see Exercise 2.12).

      Exercise 2.15: Eva Lu Ator, another user, has also noticed
      the different intervals computed by different but algebraically
      equivalent expressions. She says that a formula to compute
      with intervals using Alyssa’s system will produce tighter
      error bounds if it can be wrien in such a form that no vari-
      able that represents an uncertain number is repeated. us,
      she says, par2 is a “beer” program for parallel resistances
      than par1. Is she right? Why?

      Exercise 2.16: Explain, in general, why equivalent alge-
      braic expressions may lead to different answers. Can you
      devise an interval-arithmetic package that does not have
      this shortcoming, or is this task impossible? (Warning: is
      problem is very difficult.)

2.2 Hierarchical Data and the Closure Property
As we have seen, pairs provide a primitive “glue” that we can use to
construct compound data objects. Figure 2.2 shows a standard way to
visualize a pair—in this case, the pair formed by (cons 1 2). In this
representation, which is called box-and-pointer notation, each object is
shown as a pointer to a box. e box for a primitive object contains a
representation of the object. For example, the box for a number contains
a numeral. e box for a pair is actually a double box, the le part con-
taining (a pointer to) the car of the pair and the right part containing
the cdr.
    We have already seen that cons can be used to combine not only
numbers but pairs as well. (You made use of this fact, or should have,
in doing Exercise 2.2 and Exercise 2.3.) As a consequence, pairs pro-
vide a universal building block from which we can construct all sorts of



      Figure 2.2: Box-and-pointer representation of (cons 1 2).


                           3     4

           1     2                                 1            2       3

         (cons (cons 1 2)                       (cons (cons 1
               (cons 3 4))                                  (cons 2 3))

      Figure 2.3: Two ways to combine 1, 2, 3, and 4 using pairs.

data structures. Figure 2.3 shows two ways to use pairs to combine the
numbers 1, 2, 3, and 4.
     e ability to create pairs whose elements are pairs is the essence
of list structure’s importance as a representational tool. We refer to this
ability as the closure property of cons. In general, an operation for com-
bining data objects satisfies the closure property if the results of com-
bining things with that operation can themselves be combined using the
same operation.6 Closure is the key to power in any means of combina-
  6 e   use of the word “closure” here comes from abstract algebra, where a set of

tion because it permits us to create hierarchical structures—structures
made up of parts, which themselves are made up of parts, and so on.
    From the outset of Chapter 1, we’ve made essential use of closure
in dealing with procedures, because all but the very simplest programs
rely on the fact that the elements of a combination can themselves be
combinations. In this section, we take up the consequences of closure
for compound data. We describe some conventional techniques for us-
ing pairs to represent sequences and trees, and we exhibit a graphics
language that illustrates closure in a vivid way.7

2.2.1 Representing Sequences
One of the useful structures we can build with pairs is a sequence—an
ordered collection of data objects. ere are, of course, many ways to
elements is said to be closed under an operation if applying the operation to elements
in the set produces an element that is again an element of the set. e Lisp community
also (unfortunately) uses the word “closure” to describe a totally unrelated concept: A
closure is an implementation technique for representing procedures with free variables.
We do not use the word “closure” in this second sense in this book.
    7 e notion that a means of combination should satisfy closure is a straightfor-

ward idea. Unfortunately, the data combiners provided in many popular programming
languages do not satisfy closure, or make closure cumbersome to exploit. In Fortran
or Basic, one typically combines data elements by assembling them into arrays—but
one cannot form arrays whose elements are themselves arrays. Pascal and C admit
structures whose elements are structures. However, this requires that the program-
mer manipulate pointers explicitly, and adhere to the restriction that each field of a
structure can contain only elements of a prespecified form. Unlike Lisp with its pairs,
these languages have no built-in general-purpose glue that makes it easy to manipulate
compound data in a uniform way. is limitation lies behind Alan Perlis’s comment in
his foreword to this book: “In Pascal the plethora of declarable data structures induces
a specialization within functions that inhibits and penalizes casual cooperation. It is
beer to have 100 functions operate on one data structure than to have 10 functions
operate on 10 data structures.”

                         1            2              3              4

          Figure 2.4: e sequence 1, 2, 3, 4 represented as a chain
          of pairs.

represent sequences in terms of pairs. One particularly straightforward
representation is illustrated in Figure 2.4, where the sequence 1, 2, 3, 4 is
represented as a chain of pairs. e car of each pair is the corresponding
item in the chain, and the cdr of the pair is the next pair in the chain.
e cdr of the final pair signals the end of the sequence by pointing to
a distinguished value that is not a pair, represented in box-and-pointer
diagrams as a diagonal line and in programs as the value of the variable
nil. e entire sequence is constructed by nested cons operations:

(cons 1
          (cons 2
                  (cons 3
                             (cons 4 nil))))

     Such a sequence of pairs, formed by nested conses, is called a list,
and Scheme provides a primitive called list to help in constructing
lists.8 e above sequence could be produced by (list 1 2 3 4). In
(list     ⟨a1 ⟩ ⟨a2 ⟩ . . . ⟨an ⟩)
   8 In this book, we use list to mean a chain of pairs terminated by the end-of-list
marker. In contrast, the term list structure refers to any data structure made out of pairs,
not just to lists.

is equivalent to
(cons   ⟨a1 ⟩
        (cons   ⟨a2 ⟩
                (cons   ...
                        (cons   ⟨an ⟩
                                nil). . .)))

Lisp systems conventionally print lists by printing the sequence of el-
ements, enclosed in parentheses. us, the data object in Figure 2.4 is
printed as (1 2 3 4):
(define one-through-four (list 1 2 3 4))
(1 2 3 4)

Be careful not to confuse the expression (list 1 2 3 4) with the list
(1 2 3 4), which is the result obtained when the expression is evalu-
ated. Aempting to evaluate the expression (1 2 3 4) will signal an
error when the interpreter tries to apply the procedure 1 to arguments
2, 3, and 4.
    We can think of car as selecting the first item in the list, and of
cdr as selecting the sublist consisting of all but the first item. Nested
applications of car and cdr can be used to extract the second, third,
and subsequent items in the list.9 e constructor cons makes a list like
the original one, but with an additional item at the beginning.
   9 Sincenested applications of car and cdr are cumbersome to write, Lisp dialects
provide abbreviations for them—for instance,
(cadr   ⟨arg⟩) = (car (cdr ⟨arg⟩))
   e names of all such procedures start with c and end with r. Each a between them
stands for a car operation and each d for a cdr operation, to be applied in the same
order in which they appear in the name. e names car and cdr persist because simple
combinations like cadr are pronounceable.

(car one-through-four)
(cdr one-through-four)
(2 3 4)
(car (cdr one-through-four))
(cons 10 one-through-four)
(10 1 2 3 4)
(cons 5 one-through-four)
(5 1 2 3 4)

e value of nil, used to terminate the chain of pairs, can be thought of
as a sequence of no elements, the empty list. e word nil is a contraction
of the Latin word nihil, which means “nothing.”10

List operations
e use of pairs to represent sequences of elements as lists is accompa-
nied by conventional programming techniques for manipulating lists by
successively “cdring down” the lists. For example, the procedure list-
ref takes as arguments a list and a number n and returns the n th item
of the list. It is customary to number the elements of the list beginning
with 0. e method for computing list-ref is the following:
   10 It’s remarkable how much energy in the standardization of Lisp dialects has been

dissipated in arguments that are literally over nothing: Should nil be an ordinary
name? Should the value of nil be a symbol? Should it be a list? Should it be a pair?
In Scheme, nil is an ordinary name, which we use in this section as a variable whose
value is the end-of-list marker (just as true is an ordinary variable that has a true value).
Other dialects of Lisp, including Common Lisp, treat nil as a special symbol. e au-
thors of this book, who have endured too many language standardization brawls, would
like to avoid the entire issue. Once we have introduced quotation in Section 2.3, we will
denote the empty list as '() and dispense with the variable nil entirely.

     • For n = 0, list-ref should return the car of the list.

     • Otherwise, list-ref should return the (n − 1)-st item of the cdr
       of the list.

(define (list-ref items n)
    (if (= n 0)
        (car items)
        (list-ref (cdr items) (- n 1))))
(define squares (list 1 4 9 16 25))
(list-ref squares 3)

Oen we cdr down the whole list. To aid in this, Scheme includes a
primitive predicate null?, which tests whether its argument is the empty
list. e procedure length, which returns the number of items in a list,
illustrates this typical paern of use:
(define (length items)
    (if (null? items)
        (+ 1 (length (cdr items)))))
(define odds (list 1 3 5 7))
(length odds)

e length procedure implements a simple recursive plan. e reduc-
tion step is:

     • e length of any list is 1 plus the length of the cdr of the list.

is is applied successively until we reach the base case:

     • e length of the empty list is 0.

We could also compute length in an iterative style:
(define (length items)
  (define (length-iter a count)
    (if (null? a)
        (length-iter (cdr a) (+ 1 count))))
  (length-iter items 0))

Another conventional programming technique is to “cons up” an an-
swer list while cdring down a list, as in the procedure append, which
takes two lists as arguments and combines their elements to make a new
(append squares odds)
(1 4 9 16 25 1 3 5 7)
(append odds squares)
(1 3 5 7 1 4 9 16 25)

append is also implemented using a recursive plan. To append lists list1
and list2, do the following:

    • If list1 is the empty list, then the result is just list2.

    • Otherwise, append the cdr of list1 and list2, and cons the car
      of list1 onto the result:

(define (append list1 list2)
  (if (null? list1)
      (cons (car list1) (append (cdr list1) list2))))

      Exercise 2.17: Define a procedure last-pair that returns
      the list that contains only the last element of a given (nonempty)

(last-pair (list 23 72 149 34))

Exercise 2.18: Define a procedure reverse that takes a list
as argument and returns a list of the same elements in re-
verse order:
(reverse (list 1 4 9 16 25))
(25 16 9 4 1)

Exercise 2.19: Consider the change-counting program of
Section 1.2.2. It would be nice to be able to easily change the
currency used by the program, so that we could compute
the number of ways to change a British pound, for example.
As the program is wrien, the knowledge of the currency is
distributed partly into the procedure first-denomination
and partly into the procedure count-change (which knows
that there are five kinds of U.S. coins). It would be nicer
to be able to supply a list of coins to be used for making
We want to rewrite the procedure cc so that its second ar-
gument is a list of the values of the coins to use rather than
an integer specifying which coins to use. We could then
have lists that defined each kind of currency:
(define us-coins (list 50 25 10 5 1))
(define uk-coins (list 100 50 20 10 5 2 1 0.5))

We could then call cc as follows:
(cc 100 us-coins)

To do this will require changing the program cc somewhat.
It will still have the same form, but it will access its second
argument differently, as follows:
(define (cc amount coin-values)
  (cond ((= amount 0) 1)
         ((or (< amount 0) (no-more? coin-values)) 0)
          (+ (cc amount
              (cc (- amount

Define the procedures first-denomination, except-first-
denomination, and no-more? in terms of primitive oper-
ations on list structures. Does the order of the list coin-
values affect the answer produced by cc? Why or why not?

Exercise 2.20: e procedures +, *, and list take arbitrary
numbers of arguments. One way to define such procedures
is to use define with doed-tail notation. In a procedure
definition, a parameter list that has a dot before the last pa-
rameter name indicates that, when the procedure is called,
the initial parameters (if any) will have as values the initial
arguments, as usual, but the final parameter’s value will be
a list of any remaining arguments. For instance, given the
(define (f x y . z)    ⟨body⟩)

          the procedure f can be called with two or more arguments.
          If we evaluate
          (f 1 2 3 4 5 6)

          then in the body of f, x will be 1, y will be 2, and z will be
          the list (3 4 5 6). Given the definition
          (define (g . w)    ⟨body⟩)

          the procedure g can be called with zero or more arguments.
          If we evaluate
          (g 1 2 3 4 5 6)

          then in the body of g, w will be the list (1 2 3 4 5 6).11
          Use this notation to write a procedure same-parity that
          takes one or more integers and returns a list of all the ar-
          guments that have the same even-odd parity as the first
          argument. For example,
          (same-parity 1 2 3 4 5 6 7)
          (1 3 5 7)
          (same-parity 2 3 4 5 6 7)
          (2 4 6)

  11 To   define f and g using lambda we would write
(define f (lambda (x y . z)         ⟨body⟩))
(define g (lambda w      ⟨body⟩))

Mapping over lists
One extremely useful operation is to apply some transformation to each
element in a list and generate the list of results. For instance, the follow-
ing procedure scales each number in a list by a given factor:
(define (scale-list items factor)
  (if (null? items)
        (cons (* (car items) factor)
                (scale-list (cdr items)
(scale-list (list 1 2 3 4 5) 10)
(10 20 30 40 50)

We can abstract this general idea and capture it as a common paern
expressed as a higher-order procedure, just as in Section 1.3. e higher-
order procedure here is called map. map takes as arguments a procedure
of one argument and a list, and returns a list of the results produced by
applying the procedure to each element in the list:12
(define (map proc items)
  (if (null? items)

  12  Scheme standardly provides a map procedure that is more general than the one
described here. is more general map takes a procedure of n arguments, together with
n lists, and applies the procedure to all the first elements of the lists, all the second
elements of the lists, and so on, returning a list of the results. For example:
(map + (list 1 2 3) (list 40 50 60) (list 700 800 900))
(741 852 963)
(map (lambda (x y) (+ x (* 2 y)))
       (list 1 2 3)
       (list 4 5 6))
(9 12 15)

      (cons (proc (car items))
             (map proc (cdr items)))))
(map abs (list -10 2.5 -11.6 17))
(10 2.5 11.6 17)
(map (lambda (x) (* x x)) (list 1 2 3 4))
(1 4 9 16)

Now we can give a new definition of scale-list in terms of map:
(define (scale-list items factor)
  (map (lambda (x) (* x factor))

map   is an important construct, not only because it captures a common
paern, but because it establishes a higher level of abstraction in dealing
with lists. In the original definition of scale-list, the recursive struc-
ture of the program draws aention to the element-by-element process-
ing of the list. Defining scale-list in terms of map suppresses that level
of detail and emphasizes that scaling transforms a list of elements to a
list of results. e difference between the two definitions is not that the
computer is performing a different process (it isn’t) but that we think
about the process differently. In effect, map helps establish an abstrac-
tion barrier that isolates the implementation of procedures that trans-
form lists from the details of how the elements of the list are extracted
and combined. Like the barriers shown in Figure 2.1, this abstraction
gives us the flexibility to change the low-level details of how sequences
are implemented, while preserving the conceptual framework of oper-
ations that transform sequences to sequences. Section 2.2.3 expands on
this use of sequences as a framework for organizing programs.

      Exercise 2.21: e procedure square-list takes a list of
      numbers as argument and returns a list of the squares of

those numbers.
(square-list (list 1 2 3 4))
(1 4 9 16)

Here are two different definitions of square-list. Com-
plete both of them by filling in the missing expressions:
(define (square-list items)
  (if (null? items)
      (cons   ⟨??⟩ ⟨??⟩)))
(define (square-list items)
  (map   ⟨??⟩ ⟨??⟩))

Exercise 2.22: Louis Reasoner tries to rewrite the first square-
list procedure of Exercise 2.21 so that it evolves an itera-
tive process:
(define (square-list items)
  (define (iter things answer)
    (if (null? things)
          (iter (cdr things)
                   (cons (square (car things))
  (iter items nil))

Unfortunately, defining square-list this way produces the
answer list in the reverse order of the one desired. Why?
Louis then tries to fix his bug by interchanging the argu-
ments to cons:
(define (square-list items)

     (define (iter things answer)
      (if (null? things)
          (iter (cdr things)
                   (cons answer
                         (square (car things))))))
     (iter items nil))

is doesn’t work either. Explain.

Exercise 2.23: e procedure for-each is similar to map. It
takes as arguments a procedure and a list of elements. How-
ever, rather than forming a list of the results, for-each just
applies the procedure to each of the elements in turn, from
le to right. e values returned by applying the procedure
to the elements are not used at all—for-each is used with
procedures that perform an action, such as printing. For ex-
(for-each (lambda (x)
               (display x))
            (list 57 321 88))

e value returned by the call to for-each (not illustrated
above) can be something arbitrary, such as true. Give an
implementation of for-each.

                                                    (3 4)

       ((1 2) 3 4)

              (1 2)
                                                    3            4

                        1            2

       Figure 2.5: Structure formed by (cons (list 1 2) (list
       3 4)).

2.2.2 Hierarchical Structures
e representation of sequences in terms of lists generalizes naturally
to represent sequences whose elements may themselves be sequences.
For example, we can regard the object ((1 2) 3 4) constructed by
(cons (list 1 2) (list 3 4))

as a list of three items, the first of which is itself a list, (1 2). Indeed, this
is suggested by the form in which the result is printed by the interpreter.
Figure 2.5 shows the representation of this structure in terms of pairs.
     Another way to think of sequences whose elements are sequences
is as trees. e elements of the sequence are the branches of the tree, and
elements that are themselves sequences are subtrees. Figure 2.6 shows
the structure in Figure 2.5 viewed as a tree.
     Recursion is a natural tool for dealing with tree structures, since we
can oen reduce operations on trees to operations on their branches,
which reduce in turn to operations on the branches of the branches, and
so on, until we reach the leaves of the tree. As an example, compare the

                                ((1 2) 3 4)

                             (1 2)
                                           3   4

                               1       2

      Figure 2.6: e list structure in Figure 2.5 viewed as a tree.

length procedure of Section 2.2.1 with the count-leaves procedure,
which returns the total number of leaves of a tree:
(define x (cons (list 1 2) (list 3 4)))
(length x)
(count-leaves x)
(list x x)
(((1 2) 3 4) ((1 2) 3 4))
(length (list x x))
(count-leaves (list x x))

To implement count-leaves, recall the recursive plan for computing

    • length of a list x is 1 plus length of the cdr of x.

    • length of the empty list is 0.

count-leaves   is similar. e value for the empty list is the same:

    • count-leaves of the empty list is 0.

But in the reduction step, where we strip off the car of the list, we must
take into account that the car may itself be a tree whose leaves we need
to count. us, the appropriate reduction step is

    • count-leaves of a tree x is count-leaves of the car of x plus
      count-leaves of the cdr of x.

Finally, by taking cars we reach actual leaves, so we need another base

    • count-leaves of a leaf is 1.

To aid in writing recursive procedures on trees, Scheme provides the
primitive predicate pair?, which tests whether its argument is a pair.
Here is the complete procedure:13
(define (count-leaves x)
  (cond ((null? x) 0)
            ((not (pair? x)) 1)
            (else (+ (count-leaves (car x))
                         (count-leaves (cdr x))))))

        Exercise 2.24: Suppose we evaluate the expression (list
        1 (list 2 (list 3 4))).      Give the result printed by the
        interpreter, the corresponding box-and-pointer structure,
        and the interpretation of this as a tree (as in Figure 2.6).

        Exercise 2.25: Give combinations of cars and cdrs that
        will pick 7 from each of the following lists:
  13 e order of the first two clauses in the cond maers, since the empty list satisfies

null?   and also is not a pair.

(1 3 (5 7) 9)
(1 (2 (3 (4 (5 (6 7))))))

Exercise 2.26: Suppose we define x and y to be two lists:
(define x (list 1 2 3))
(define y (list 4 5 6))

What result is printed by the interpreter in response to eval-
uating each of the following expressions:
(append x y)
(cons x y)
(list x y)

Exercise 2.27: Modify your reverse procedure of Exercise
2.18 to produce a deep-reverse procedure that takes a list
as argument and returns as its value the list with its ele-
ments reversed and with all sublists deep-reversed as well.
For example,
(define x (list (list 1 2) (list 3 4)))
((1 2) (3 4))
(reverse x)
((3 4) (1 2))
(deep-reverse x)
((4 3) (2 1))

Exercise 2.28: Write a procedure fringe that takes as argu-
ment a tree (represented as a list) and returns a list whose
elements are all the leaves of the tree arranged in le-to-
right order. For example,

(define x (list (list 1 2) (list 3 4)))
(fringe x)
(1 2 3 4)
(fringe (list x x))
(1 2 3 4 1 2 3 4)

Exercise 2.29: A binary mobile consists of two branches,
a le branch and a right branch. Each branch is a rod of
a certain length, from which hangs either a weight or an-
other binary mobile. We can represent a binary mobile us-
ing compound data by constructing it from two branches
(for example, using list):
(define (make-mobile left right)
  (list left right))

A branch is constructed from a length (which must be a
number) together with a structure, which may be either a
number (representing a simple weight) or another mobile:
(define (make-branch length structure)
  (list length structure))

  a. Write the corresponding selectors left-branch and
     right-branch, which return the branches of a mobile,
     and branch-length and branch-structure, which re-
     turn the components of a branch.
  b. Using your selectors, define a procedure total-weight
     that returns the total weight of a mobile.
  c. A mobile is said to be balanced if the torque applied by
     its top-le branch is equal to that applied by its top-

           right branch (that is, if the length of the le rod mul-
           tiplied by the weight hanging from that rod is equal
           to the corresponding product for the right side) and if
           each of the submobiles hanging off its branches is bal-
           anced. Design a predicate that tests whether a binary
           mobile is balanced.
        d. Suppose we change the representation of mobiles so
           that the constructors are
           (define (make-mobile left right) (cons left right))
           (define (make-branch length structure)
              (cons length structure))

           How much do you need to change your programs to
           convert to the new representation?

Mapping over trees
Just as map is a powerful abstraction for dealing with sequences, map
together with recursion is a powerful abstraction for dealing with trees.
For instance, the scale-tree procedure, analogous to scale-list of
Section 2.2.1, takes as arguments a numeric factor and a tree whose
leaves are numbers. It returns a tree of the same shape, where each
number is multiplied by the factor. e recursive plan for scale-tree
is similar to the one for count-leaves:
(define (scale-tree tree factor)
  (cond ((null? tree) nil)
         ((not (pair? tree)) (* tree factor))
         (else (cons (scale-tree (car tree) factor)
                      (scale-tree (cdr tree) factor)))))
(scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7)) 10)
(10 (20 (30 40) 50) (60 70))

Another way to implement scale-tree is to regard the tree as a se-
quence of sub-trees and use map. We map over the sequence, scaling
each sub-tree in turn, and return the list of results. In the base case,
where the tree is a leaf, we simply multiply by the factor:
(define (scale-tree tree factor)
  (map (lambda (sub-tree)
          (if (pair? sub-tree)
              (scale-tree sub-tree factor)
              (* sub-tree factor)))

Many tree operations can be implemented by similar combinations of
sequence operations and recursion.

      Exercise 2.30: Define a procedure square-tree analogous
      to the square-list procedure of Exercise 2.21. at is, square-
      tree should behave as follows:

       (list 1
              (list 2 (list 3 4) 5)
              (list 6 7)))
      (1 (4 (9 16) 25) (36 49))

      Define square-tree both directly (i.e., without using any
      higher-order procedures) and also by using map and recur-

      Exercise 2.31: Abstract your answer to Exercise 2.30 to
      produce a procedure tree-map with the property that square-
      tree could be defined as

      (define (square-tree tree) (tree-map square tree))

      Exercise 2.32: We can represent a set as a list of distinct
      elements, and we can represent the set of all subsets of the
      set as a list of lists. For example, if the set is (1 2 3), then
      the set of all subsets is (() (3) (2) (2 3) (1) (1 3)
      (1 2) (1 2 3)). Complete the following definition of a
      procedure that generates the set of subsets of a set and give
      a clear explanation of why it works:
      (define (subsets s)
        (if (null? s)
             (list nil)
             (let ((rest (subsets (cdr s))))
               (append rest (map     ⟨??⟩ rest)))))

2.2.3 Sequences as Conventional Interfaces
In working with compound data, we’ve stressed how data abstraction
permits us to design programs without becoming enmeshed in the de-
tails of data representations, and how abstraction preserves for us the
flexibility to experiment with alternative representations. In this sec-
tion, we introduce another powerful design principle for working with
data structures—the use of conventional interfaces.
     In Section 1.3 we saw how program abstractions, implemented as
higher-order procedures, can capture common paerns in programs
that deal with numerical data. Our ability to formulate analogous oper-
ations for working with compound data depends crucially on the style
in which we manipulate our data structures. Consider, for example, the
following procedure, analogous to the count-leaves procedure of Sec-
tion 2.2.2, which takes a tree as argument and computes the sum of the
squares of the leaves that are odd:

(define (sum-odd-squares tree)
  (cond ((null? tree) 0)
        ((not (pair? tree))
          (if (odd? tree) (square tree) 0))
        (else (+ (sum-odd-squares (car tree))
                  (sum-odd-squares (cdr tree))))))

On the surface, this procedure is very different from the following one,
which constructs a list of all the even Fibonacci numbers Fib(k), where
k is less than or equal to a given integer n:
(define (even-fibs n)
  (define (next k)
    (if (> k n)
        (let ((f (fib k)))
           (if (even? f)
               (cons f (next (+ k 1)))
               (next (+ k 1))))))
  (next 0))

Despite the fact that these two procedures are structurally very differ-
ent, a more abstract description of the two computations reveals a great
deal of similarity. e first program

    • enumerates the leaves of a tree;

    • filters them, selecting the odd ones;

    • squares each of the selected ones; and

    • accumulates the results using +, starting with 0.

e second program

 enumerate:         filter:             map:             accumulate:
 tree leaves        odd?                square           +, 0

 enumerate:         map:                filter:          accumulate:
 integers           fib                 even?            cons, ()

      Figure 2.7: e signal-flow plans for the procedures sum-
      odd-squares (top) and even-fibs (boom) reveal the com-
      monality between the two programs.

    • enumerates the integers from 0 to n;

    • computes the Fibonacci number for each integer;

    • filters them, selecting the even ones; and

    • accumulates the results using cons, starting with the empty list.

A signal-processing engineer would find it natural to conceptualize these
processes in terms of signals flowing through a cascade of stages, each
of which implements part of the program plan, as shown in Figure 2.7.
In sum-odd-squares, we begin with an enumerator, which generates a
“signal” consisting of the leaves of a given tree. is signal is passed
through a filter, which eliminates all but the odd elements. e result-
ing signal is in turn passed through a map, which is a “transducer” that
applies the square procedure to each element. e output of the map
is then fed to an accumulator, which combines the elements using +,
starting from an initial 0. e plan for even-fibs is analogous.
    Unfortunately, the two procedure definitions above fail to exhibit
this signal-flow structure. For instance, if we examine the sum-odd-

squares procedure, we find that the enumeration is implemented partly
by the null? and pair? tests and partly by the tree-recursive structure
of the procedure. Similarly, the accumulation is found partly in the tests
and partly in the addition used in the recursion. In general, there are no
distinct parts of either procedure that correspond to the elements in the
signal-flow description. Our two procedures decompose the computa-
tions in a different way, spreading the enumeration over the program
and mingling it with the map, the filter, and the accumulation. If we
could organize our programs to make the signal-flow structure manifest
in the procedures we write, this would increase the conceptual clarity
of the resulting code.

Sequence Operations
e key to organizing programs so as to more clearly reflect the signal-
flow structure is to concentrate on the “signals” that flow from one stage
in the process to the next. If we represent these signals as lists, then we
can use list operations to implement the processing at each of the stages.
For instance, we can implement the mapping stages of the signal-flow
diagrams using the map procedure from Section 2.2.1:
(map square (list 1 2 3 4 5))
(1 4 9 16 25)

Filtering a sequence to select only those elements that satisfy a given
predicate is accomplished by
(define (filter predicate sequence)
  (cond ((null? sequence) nil)
         ((predicate (car sequence))
          (cons (car sequence)
                 (filter predicate (cdr sequence))))
         (else (filter predicate (cdr sequence)))))

For example,
(filter odd? (list 1 2 3 4 5))
(1 3 5)

Accumulations can be implemented by
(define (accumulate op initial sequence)
  (if (null? sequence)
       (op (car sequence)
             (accumulate op initial (cdr sequence)))))
(accumulate + 0 (list 1 2 3 4 5))
(accumulate * 1 (list 1 2 3 4 5))
(accumulate cons nil (list 1 2 3 4 5))
(1 2 3 4 5)

All that remains to implement signal-flow diagrams is to enumerate the
sequence of elements to be processed. For even-fibs, we need to gen-
erate the sequence of integers in a given range, which we can do as
(define (enumerate-interval low high)
  (if (> low high)
       (cons low (enumerate-interval (+ low 1) high))))
(enumerate-interval 2 7)
(2 3 4 5 6 7)

To enumerate the leaves of a tree, we can use14
  14 isis, in fact, precisely the fringe procedure from Exercise 2.28. Here we’ve re-
named it to emphasize that it is part of a family of general sequence-manipulation

(define (enumerate-tree tree)
  (cond ((null? tree) nil)
          ((not (pair? tree)) (list tree))
          (else (append (enumerate-tree (car tree))
                        (enumerate-tree (cdr tree))))))
(enumerate-tree (list 1 (list 2 (list 3 4)) 5))
(1 2 3 4 5)

Now we can reformulate sum-odd-squares and even-fibs as in the
signal-flow diagrams. For sum-odd-squares, we enumerate the sequence
of leaves of the tree, filter this to keep only the odd numbers in the se-
quence, square each element, and sum the results:
(define (sum-odd-squares tree)
   + 0 (map square (filter odd? (enumerate-tree tree)))))

For even-fibs, we enumerate the integers from 0 to n, generate the Fi-
bonacci number for each of these integers, filter the resulting sequence
to keep only the even elements, and accumulate the results into a list:
(define (even-fibs n)
   (filter even? (map fib (enumerate-interval 0 n)))))

e value of expressing programs as sequence operations is that this
helps us make program designs that are modular, that is, designs that
are constructed by combining relatively independent pieces. We can en-
courage modular design by providing a library of standard components
together with a conventional interface for connecting the components
in flexible ways.

    Modular construction is a powerful strategy for controlling com-
plexity in engineering design. In real signal-processing applications, for
example, designers regularly build systems by cascading elements se-
lected from standardized families of filters and transducers. Similarly,
sequence operations provide a library of standard program elements
that we can mix and match. For instance, we can reuse pieces from the
sum-odd-squares and even-fibs procedures in a program that con-
structs a list of the squares of the first n + 1 Fibonacci numbers:
(define (list-fib-squares n)
   (map square (map fib (enumerate-interval 0 n)))))
(list-fib-squares 10)
(0 1 1 4 9 25 64 169 441 1156 3025)

We can rearrange the pieces and use them in computing the product of
the squares of the odd integers in a sequence:
(define (product-of-squares-of-odd-elements sequence)
  (accumulate * 1 (map square (filter odd? sequence))))
(product-of-squares-of-odd-elements (list 1 2 3 4 5))

We can also formulate conventional data-processing applications in terms
of sequence operations. Suppose we have a sequence of personnel records
and we want to find the salary of the highest-paid programmer. Assume
that we have a selector salary that returns the salary of a record, and a
predicate programmer? that tests if a record is for a programmer. en
we can write
(define (salary-of-highest-paid-programmer records)
  (accumulate max 0 (map salary (filter programmer? records))))

ese examples give just a hint of the vast range of operations that can
be expressed as sequence operations.15
    Sequences, implemented here as lists, serve as a conventional in-
terface that permits us to combine processing modules. Additionally,
when we uniformly represent structures as sequences, we have local-
ized the data-structure dependencies in our programs to a small number
of sequence operations. By changing these, we can experiment with al-
ternative representations of sequences, while leaving the overall design
of our programs intact. We will exploit this capability in Section 3.5,
when we generalize the sequence-processing paradigm to admit infi-
nite sequences.

       Exercise 2.33: Fill in the missing expressions to complete
       the following definitions of some basic list-manipulation
       operations as accumulations:
       (define (map p sequence)
          (accumulate (lambda (x y)           ⟨??⟩) nil sequence))
       (define (append seq1 seq2)
          (accumulate cons        ⟨??⟩ ⟨??⟩))
       (define (length sequence)
          (accumulate      ⟨??⟩ 0 sequence))

     Richard Waters (1979) developed a program that automatically analyzes traditional
Fortran programs, viewing them in terms of maps, filters, and accumulations. He found
that fully 90 percent of the code in the Fortran Scientific Subroutine Package fits neatly
into this paradigm. One of the reasons for the success of Lisp as a programming lan-
guage is that lists provide a standard medium for expressing ordered collections so that
they can be manipulated using higher-order operations. e programming language
APL owes much of its power and appeal to a similar choice. In APL all data are repre-
sented as arrays, and there is a universal and convenient set of generic operators for all
sorts of array operations.

        Exercise 2.34: Evaluating a polynomial in x at a given value
        of x can be formulated as an accumulation. We evaluate the

                        an x n + an −1x n −1 + · · · + a 1x + a 0

        using a well-known algorithm called Horner’s rule, which
        structures the computation as

                       (. . . (an x + an −1 )x + · · · + a 1 )x + a 0 .

        In other words, we start with an , multiply by x, add an −1 ,
        multiply by x, and so on, until we reach a 0 .16
        Fill in the following template to produce a procedure that
        evaluates a polynomial using Horner’s rule. Assume that
        the coefficients of the polynomial are arranged in a sequence,
        from a 0 through an .
        (define (horner-eval x coefficient-sequence)
          (accumulate (lambda (this-coeff higher-terms)                     ⟨??⟩)

   16 According to Knuth 1981, this rule was formulated by W. G. Horner early in the

nineteenth century, but the method was actually used by Newton over a hundred years
earlier. Horner’s rule evaluates the polynomial using fewer additions and multipli-
cations than does the straightforward method of first computing an x n , then adding
an −1 x n −1 , and so on. In fact, it is possible to prove that any algorithm for evaluating
arbitrary polynomials must use at least as many additions and multiplications as does
Horner’s rule, and thus Horner’s rule is an optimal algorithm for polynomial evaluation.
is was proved (for the number of additions) by A. M. Ostrowski in a 1954 paper that
essentially founded the modern study of optimal algorithms. e analogous statement
for multiplications was proved by V. Y. Pan in 1966. e book by Borodin and Munro
(1975) provides an overview of these and other results about optimal algorithms.

For example, to compute 1+3x +5x 3 +x 5 at x = 2 you would
(horner-eval 2 (list 1 3 0 5 0 1))

Exercise 2.35: Redefine count-leaves from Section 2.2.2
as an accumulation:
(define (count-leaves t)
  (accumulate   ⟨??⟩ ⟨??⟩ (map ⟨??⟩ ⟨??⟩)))

Exercise 2.36: e procedure accumulate-n is similar to
accumulate   except that it takes as its third argument a se-
quence of sequences, which are all assumed to have the
same number of elements. It applies the designated accu-
mulation procedure to combine all the first elements of the
sequences, all the second elements of the sequences, and so
on, and returns a sequence of the results. For instance, if s
is a sequence containing four sequences, ((1 2 3) (4 5 6)
(7 8 9) (10 11 12)), then the value of (accumulate-n +
0 s) should be the sequence (22 26 30). Fill in the missing
expressions in the following definition of accumulate-n:
(define (accumulate-n op init seqs)
  (if (null? (car seqs))
      (cons (accumulate op init     ⟨??⟩)
             (accumulate-n op init    ⟨??⟩))))

Exercise 2.37: Suppose we represent vectors v = (vi ) as
sequences of numbers, and matrices m = (mij ) as sequences

    of vectors (the rows of the matrix). For example, the matrix
                                                    
                                  1 2 3 4 
                                                  
                                      4 5 6 6 
                                         6 7 8 9

    is represented as the sequence ((1 2 3 4) (4 5 6 6)
    (6 7 8 9)). With this representation, we can use sequence
    operations to concisely express the basic matrix and vector
    operations. ese operations (which are described in any
    book on matrix algebra) are the following:

                                 returns the sum Σi vi wi ;
                 (dot-product v w)
           (matrix-*-vector m v) returns the vector t,
                                 where ti = Σj mij vj ;
           (matrix-*-matrix m n) returns the matrix p ,
                                 where pij = Σk mik n kj ;
                   (transpose m) returns the matrix n ,
                                 where nij = mji .

    We can define the dot product as17
    (define (dot-product v w)
         (accumulate + 0 (map * v w)))

    Fill in the missing expressions in the following procedures
    for computing the other matrix operations. (e procedure
    accumulate-n is defined in Exercise 2.36.)

    (define (matrix-*-vector m v)
         (map   ⟨??⟩ m))
17 is   definition uses the extended version of map described in Footnote 12.

(define (transpose mat)
  (accumulate-n   ⟨??⟩ ⟨??⟩ mat))
(define (matrix-*-matrix m n)
  (let ((cols (transpose n)))
    (map   ⟨??⟩ m)))

Exercise 2.38: e accumulate procedure is also known as
fold-right, because it combines the first element of the se-
quence with the result of combining all the elements to the
right. ere is also a fold-left, which is similar to fold-
right, except that it combines elements working in the op-
posite direction:
(define (fold-left op initial sequence)
  (define (iter result rest)
    (if (null? rest)
        (iter (op result (car rest))
                 (cdr rest))))
  (iter initial sequence))

What are the values of
(fold-right / 1 (list 1 2 3))
(fold-left / 1 (list 1 2 3))
(fold-right list nil (list 1 2 3))
(fold-left list nil (list 1 2 3))

Give a property that op should satisfy to guarantee that
fold-right and fold-left will produce the same values
for any sequence.

       Exercise 2.39: Complete the following definitions of reverse
       (Exercise 2.18) in terms of fold-right and fold-left from
       Exercise 2.38:
       (define (reverse sequence)
           (fold-right (lambda (x y)        ⟨??⟩) nil sequence))
       (define (reverse sequence)
           (fold-left (lambda (x y)        ⟨??⟩) nil sequence))

Nested Mappings
We can extend the sequence paradigm to include many computations
that are commonly expressed using nested loops.18 Consider this prob-
lem: Given a positive integer n, find all ordered pairs of distinct positive
integers i and j, where 1 ≤ j < i ≤ n, such that i + j is prime. For
example, if n is 6, then the pairs are the following:

            i         2       3       4         4     5       6       6
            j         1       2       1         3     2       1       5
           i+j        3       5       5         7     7       7       11

A natural way to organize this computation is to generate the sequence
of all ordered pairs of positive integers less than or equal to n, filter to
select those pairs whose sum is prime, and then, for each pair (i, j) that
passes through the filter, produce the triple (i, j, i + j).
    Here is a way to generate the sequence of pairs: For each integer
i ≤ n, enumerate the integers j < i, and for each such i and j gener-
ate the pair (i, j). In terms of sequence operations, we map along the
  18 is approach to nested mappings was shown to us by David Turner, whose lan-
guages KRC and Miranda provide elegant formalisms for dealing with these constructs.
e examples in this section (see also Exercise 2.42) are adapted from Turner 1981. In
Section 3.5.3, we’ll see how this approach generalizes to infinite sequences.

sequence (enumerate-interval 1 n). For each i in this sequence, we
map along the sequence (enumerate-interval 1 (- i 1)). For each
j in this laer sequence, we generate the pair (list i j). is gives
us a sequence of pairs for each i. Combining all the sequences for all
the i (by accumulating with append) produces the required sequence of
 append nil (map (lambda (i)
                         (map (lambda (j) (list i j))
                                (enumerate-interval 1 (- i 1))))
                       (enumerate-interval 1 n)))

e combination of mapping and accumulating with append is so com-
mon in this sort of program that we will isolate it as a separate proce-
(define (flatmap proc seq)
  (accumulate append nil (map proc seq)))

Now filter this sequence of pairs to find those whose sum is prime. e
filter predicate is called for each element of the sequence; its argument is
a pair and it must extract the integers from the pair. us, the predicate
to apply to each element in the sequence is
(define (prime-sum? pair)
  (prime? (+ (car pair) (cadr pair))))

Finally, generate the sequence of results by mapping over the filtered
pairs using the following procedure, which constructs a triple consisting
of the two elements of the pair along with their sum:
  19 We’rerepresenting a pair here as a list of two elements rather than as a Lisp pair.
us, the “pair” (i, j) is represented as (list i j), not (cons i j).

(define (make-pair-sum pair)
  (list (car pair) (cadr pair) (+ (car pair) (cadr pair))))

Combining all these steps yields the complete procedure:
(define (prime-sum-pairs n)
  (map make-pair-sum
          (filter prime-sum? (flatmap
                                 (lambda (i)
                                   (map (lambda (j) (list i j))
                                          (enumerate-interval 1 (- i 1))))
                                 (enumerate-interval 1 n)))))

Nested mappings are also useful for sequences other than those that
enumerate intervals. Suppose we wish to generate all the permutations
of a set S; that is, all the ways of ordering the items in the set. For in-
stance, the permutations of {1, 2, 3} are {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1},
{3, 1, 2}, and {3, 2, 1}. Here is a plan for generating the permutations of S:
For each item x in S, recursively generate the sequence of permutations
of S − x,20 and adjoin x to the front of each one. is yields, for each x
in S, the sequence of permutations of S that begin with x. Combining
these sequences for all x gives all the permutations of S:21
(define (permutations s)
  (if (null? s)                        ; empty set?
        (list nil)                     ; sequence containing empty set
        (flatmap (lambda (x)
                       (map (lambda (p) (cons x p))
                              (permutations (remove x s))))

  20 e set S − x is the set of all elements of S, excluding x.
  21 Semicolons in Scheme code are used to introduce comments. Everything from the
semicolon to the end of the line is ignored by the interpreter. In this book we don’t use
many comments; we try to make our programs self-documenting by using descriptive

Notice how this strategy reduces the problem of generating permuta-
tions of S to the problem of generating the permutations of sets with
fewer elements than S. In the terminal case, we work our way down to
the empty list, which represents a set of no elements. For this, we gen-
erate (list nil), which is a sequence with one item, namely the set
with no elements. e remove procedure used in permutations returns
all the items in a given sequence except for a given item. is can be
expressed as a simple filter:
(define (remove item sequence)
  (filter (lambda (x) (not (= x item)))

      Exercise 2.40: Define a procedure unique-pairs that, given
      an integer n, generates the sequence of pairs (i, j) with 1 ≤
      j < i ≤ n. Use unique-pairs to simplify the definition of
      prime-sum-pairs given above.

      Exercise 2.41: Write a procedure to find all ordered triples
      of distinct positive integers i, j, and k less than or equal to
      a given integer n that sum to a given integer s.

      Exercise 2.42: e “eight-queens puzzle” asks how to place
      eight queens on a chessboard so that no queen is in check
      from any other (i.e., no two queens are in the same row, col-
      umn, or diagonal). One possible solution is shown in Figure
      2.8. One way to solve the puzzle is to work across the board,
      placing a queen in each column. Once we have placed k − 1
      queens, we must place the k th queen in a position where it
      does not check any of the queens already on the board. We
      can formulate this approach recursively: Assume that we

    Figure 2.8: A solution to the eight-queens puzzle.

have already generated the sequence of all possible ways
to place k − 1 queens in the first k − 1 columns of the board.
For each of these ways, generate an extended set of posi-
tions by placing a queen in each row of the k th column.
Now filter these, keeping only the positions for which the
queen in the k th column is safe with respect to the other
queens. is produces the sequence of all ways to place k
queens in the first k columns. By continuing this process,
we will produce not only one solution, but all solutions to
the puzzle.
We implement this solution as a procedure queens, which
returns a sequence of all solutions to the problem of plac-
ing n queens on an n × n chessboard. queens has an inter-
nal procedure queen-cols that returns the sequence of all
ways to place queens in the first k columns of the board.

(define (queens board-size)
  (define (queen-cols k)
    (if (= k 0)
        (list empty-board)
         (lambda (positions) (safe? k positions))
           (lambda (rest-of-queens)
             (map (lambda (new-row)
                      new-row k rest-of-queens))
                    (enumerate-interval 1 board-size)))
           (queen-cols (- k 1))))))
  (queen-cols board-size))

In this procedure rest-of-queens is a way to place k − 1
queens in the first k −1 columns, and new-row is a proposed
row in which to place the queen for the k th column. Com-
plete the program by implementing the representation for
sets of board positions, including the procedure adjoin-
position, which adjoins a new row-column position to a
set of positions, and empty-board, which represents an empty
set of positions. You must also write the procedure safe?,
which determines for a set of positions, whether the queen
in the k th column is safe with respect to the others. (Note
that we need only check whether the new queen is safe—
the other queens are already guaranteed safe with respect
to each other.)

Exercise 2.43: Louis Reasoner is having a terrible time do-
ing Exercise 2.42. His queens procedure seems to work, but
it runs extremely slowly. (Louis never does manage to wait

       long enough for it to solve even the 6 × 6 case.) When Louis
       asks Eva Lu Ator for help, she points out that he has inter-
       changed the order of the nested mappings in the flatmap,
       writing it as
        (lambda (new-row)
           (map (lambda (rest-of-queens)
                    (adjoin-position new-row k rest-of-queens))
                  (queen-cols (- k 1))))
        (enumerate-interval 1 board-size))

       Explain why this interchange makes the program run slowly.
       Estimate how long it will take Louis’s program to solve the
       eight-queens puzzle, assuming that the program in Exercise
       2.42 solves the puzzle in time T .

2.2.4 Example: A Picture Language
is section presents a simple language for drawing pictures that il-
lustrates the power of data abstraction and closure, and also exploits
higher-order procedures in an essential way. e language is designed
to make it easy to experiment with paerns such as the ones in Fig-
ure 2.9, which are composed of repeated elements that are shied and
scaled.22 In this language, the data objects being combined are repre-
sented as procedures rather than as list structure. Just as cons, which
satisfies the closure property, allowed us to easily build arbitrarily com-
plicated list structure, the operations in this language, which also sat-
  22 e picture language is based on the language Peter Henderson created to construct

images like M.C. Escher’s “Square Limit” woodcut (see Henderson 1982). e woodcut
incorporates a repeated scaled paern, similar to the arrangements drawn using the
square-limit procedure in this section.

       Figure 2.9: Designs generated with the picture language.

isfy the closure property, allow us to easily build arbitrarily complicated

The picture language
When we began our study of programming in Section 1.1, we empha-
sized the importance of describing a language by focusing on the lan-
guage’s primitives, its means of combination, and its means of abstrac-
tion. We’ll follow that framework here.
    Part of the elegance of this picture language is that there is only one
kind of element, called a painter. A painter draws an image that is shied
and scaled to fit within a designated parallelogram-shaped frame. For
example, there’s a primitive painter we’ll call wave that makes a crude
line drawing, as shown in Figure 2.10. e actual shape of the drawing
depends on the frame—all four images in figure 2.10 are produced by the

       Figure 2.10: Images produced by the wave painter, with
       respect to four different frames. e frames, shown with
       doed lines, are not part of the images.

same wave painter, but with respect to four different frames. Painters
can be more elaborate than this: e primitive painter called rogers
paints a picture of ’s founder, William Barton Rogers, as shown in
Figure 2.11.23 e four images in figure 2.11 are drawn with respect to
  23 William  Barton Rogers (1804-1882) was the founder and first president of .
A geologist and talented teacher, he taught at William and Mary College and at the
University of Virginia. In 1859 he moved to Boston, where he had more time for re-
search, worked on a plan for establishing a “polytechnic institute,” and served as Mas-
sachuses’s first State Inspector of Gas Meters.
   When  was established in 1861, Rogers was elected its first president. Rogers
espoused an ideal of “useful learning” that was different from the university education
of the time, with its overemphasis on the classics, which, as he wrote, “stand in the way
of the broader, higher and more practical instruction and discipline of the natural and
social sciences.” is education was likewise to be different from narrow trade-school

the same four frames as the wave images in figure 2.10.
    To combine images, we use various operations that construct new
painters from given painters. For example, the beside operation takes
two painters and produces a new, compound painter that draws the first
education. In Rogers’s words:

       e world-enforced distinction between the practical and the scientific
       worker is uerly futile, and the whole experience of modern times has
       demonstrated its uer worthlessness.

   Rogers served as president of  until 1870, when he resigned due to ill health.
In 1878 the second president of , John Runkle, resigned under the pressure of a
financial crisis brought on by the Panic of 1873 and strain of fighting off aempts by
Harvard to take over . Rogers returned to hold the office of president until 1881.
   Rogers collapsed and died while addressing ’s graduating class at the commence-
ment exercises of 1882. Runkle quoted Rogers’s last words in a memorial address de-
livered that same year:

       “As I stand here today and see what the Institute is, . . . I call to mind
       the beginnings of science. I remember one hundred and fiy years ago
       Stephen Hales published a pamphlet on the subject of illuminating gas,
       in which he stated that his researches had demonstrated that 128 grains
       of bituminous coal – ” “Bituminous coal,” these were his last words on
       earth. Here he bent forward, as if consulting some notes on the table
       before him, then slowly regaining an erect position, threw up his hands,
       and was translated from the scene of his earthly labors and triumphs
       to “the tomorrow of death,” where the mysteries of life are solved, and
       the disembodied spirit finds unending satisfaction in contemplating the
       new and still unfathomable mysteries of the infinite future.

  In the words of Francis A. Walker (’s third president):

       All his life he had borne himself most faithfully and heroically, and he
       died as so good a knight would surely have wished, in harness, at his
       post, and in the very part and act of public duty.

      Figure 2.11: Images of William Barton Rogers, founder and
      first president of , painted with respect to the same four
      frames as in Figure 2.10 (original image from Wikimedia

painter’s image in the le half of the frame and the second painter’s im-
age in the right half of the frame. Similarly, below takes two painters and
produces a compound painter that draws the first painter’s image below
the second painter’s image. Some operations transform a single painter
to produce a new painter. For example, flip-vert takes a painter and
produces a painter that draws its image upside-down, and flip-horiz
produces a painter that draws the original painter’s image le-to-right
    Figure 2.12 shows the drawing of a painter called wave4 that is built
up in two stages starting from wave:
(define wave2 (beside wave (flip-vert wave)))
(define wave4 (below wave2 wave2))

      Figure 2.12: Creating a complex figure, starting from the
      wave painter of Figure 2.10.

     In building up a complex image in this manner we are exploiting the
fact that painters are closed under the language’s means of combination.
e beside or below of two painters is itself a painter; therefore, we can
use it as an element in making more complex painters. As with building
up list structure using cons, the closure of our data under the means of
combination is crucial to the ability to create complex structures while
using only a few operations.
     Once we can combine painters, we would like to be able to abstract
typical paerns of combining painters. We will implement the painter
operations as Scheme procedures. is means that we don’t need a spe-
cial abstraction mechanism in the picture language: Since the means of
combination are ordinary Scheme procedures, we automatically have
the capability to do anything with painter operations that we can do
with procedures. For example, we can abstract the paern in wave4 as
(define (flipped-pairs painter)
  (let ((painter2 (beside painter (flip-vert painter))))
    (below painter2 painter2)))

and define wave4 as an instance of this paern:
(define wave4 (flipped-pairs wave))

                                        up-     up-
                   right-split          split   split   corner-split
                       n--1             n--1     n--1        n--1


                   right-split                               n--1
                       n--1                              right-split

         right-split n                          corner-split n

  Figure 2.13: Recursive plans for right-split and corner-split.

We can also define recursive operations. Here’s one that makes painters
split and branch towards the right as shown in Figure 2.13 and Figure
(define (right-split painter n)
  (if (= n 0)
      (let ((smaller (right-split painter (- n 1))))
        (beside painter (below smaller smaller)))))

We can produce balanced paerns by branching upwards as well as
towards the right (see exercise Exercise 2.44 and figures Figure 2.13 and
Figure 2.14):
(define (corner-split painter n)
  (if (= n 0)

      (let ((up (up-split painter (- n 1)))
             (right (right-split painter (- n 1))))
        (let ((top-left (beside up up))
               (bottom-right (below right right))
               (corner (corner-split painter (- n 1))))
          (beside (below painter top-left)
                   (below bottom-right corner))))))

By placing four copies of a corner-split appropriately, we obtain a
paern called square-limit, whose application to wave and rogers is
shown in Figure 2.9:
(define (square-limit painter n)
  (let ((quarter (corner-split painter n)))
    (let ((half (beside (flip-horiz quarter) quarter)))
      (below (flip-vert half) half))))

      Exercise 2.44: Define the procedure up-split used by corner-
      split. It is similar to right-split, except that it switches
      the roles of below and beside.

Higher-order operations
In addition to abstracting paerns of combining painters, we can work
at a higher level, abstracting paerns of combining painter operations.
at is, we can view the painter operations as elements to manipulate
and can write means of combination for these elements—procedures
that take painter operations as arguments and create new painter oper-
    For example, flipped-pairs and square-limit each arrange four
copies of a painter’s image in a square paern; they differ only in how

   (right-split wave 4)          (right-split rogers 4)

  (corner-split wave 4)         (corner-split rogers 4)

Figure 2.14: e recursive operations right-split and
corner-split applied to the painters wave and rogers.
Combining four corner-split figures produces symmet-
ric square-limit designs as shown in Figure 2.9.

they orient the copies. One way to abstract this paern of painter com-
bination is with the following procedure, which takes four one-argument
painter operations and produces a painter operation that transforms a
given painter with those four operations and arranges the results in a
square. tl, tr, bl, and br are the transformations to apply to the top
le copy, the top right copy, the boom le copy, and the boom right
copy, respectively.
(define (square-of-four tl tr bl br)
  (lambda (painter)
     (let ((top (beside (tl painter) (tr painter)))
             (bottom (beside (bl painter) (br painter))))
       (below bottom top))))

en flipped-pairs can be defined in terms of square-of-four as
(define (flipped-pairs painter)
  (let ((combine4 (square-of-four identity flip-vert
                                             identity flip-vert)))
     (combine4 painter)))

and square-limit can be expressed as25
(define (square-limit painter n)
  (let ((combine4 (square-of-four flip-horiz identity
                                             rotate180 flip-vert)))

  24 Equivalently,   we could write
(define flipped-pairs
  (square-of-four identity flip-vert identity flip-vert))

  25 rotate180 rotates a painter by 180 degrees (see Exercise 2.50). Instead of ro-
tate180  we could say (compose flip-vert flip-horiz), using the compose pro-
cedure from Exercise 1.42.

    (combine4 (corner-split painter n))))

      Exercise 2.45: right-split and up-split can be expressed
      as instances of a general spliing operation. Define a pro-
      cedure split with the property that evaluating
      (define right-split (split beside below))
      (define up-split (split below beside))

      produces procedures right-split and up-split with the
      same behaviors as the ones already defined.

Before we can show how to implement painters and their means of com-
bination, we must first consider frames. A frame can be described by
three vectors—an origin vector and two edge vectors. e origin vector
specifies the offset of the frame’s origin from some absolute origin in
the plane, and the edge vectors specify the offsets of the frame’s cor-
ners from its origin. If the edges are perpendicular, the frame will be
rectangular. Otherwise the frame will be a more general parallelogram.
    Figure 2.15 shows a frame and its associated vectors. In accordance
with data abstraction, we need not be specific yet about how frames are
represented, other than to say that there is a constructor make-frame,
which takes three vectors and produces a frame, and three correspond-
ing selectors origin-frame, edge1-frame, and edge2-frame (see Exer-
cise 2.47).
    We will use coordinates in the unit square (0 ≤ x , y ≤ 1) to specify
images. With each frame, we associate a frame coordinate map, which
will be used to shi and scale images to fit the frame. e map trans-
forms the unit square into the frame by mapping the vector v = (x , y)

                       frame                            frame
                       edge2                            edge1
                       vector                           vector

                                     origin     (0, 0) point on
                                     vector     display screen

       Figure 2.15: A frame is described by three vectors — an
       origin and two edges.

to the vector sum

          Origin(Frame) + x · Edge1 (Frame) + y · Edge2 (Frame).

For example, (0, 0) is mapped to the origin of the frame, (1, 1) to the
vertex diagonally opposite the origin, and (0.5, 0.5) to the center of the
frame. We can create a frame’s coordinate map with the following pro-
(define (frame-coord-map frame)
  (lambda (v)
      (origin-frame frame)

  26 frame-coord-map    uses the vector operations described in Exercise 2.46 below,
which we assume have been implemented using some representation for vectors. Be-
cause of data abstraction, it doesn’t maer what this vector representation is, so long
as the vector operations behave correctly.

     (add-vect (scale-vect (xcor-vect v) (edge1-frame frame))
                 (scale-vect (ycor-vect v) (edge2-frame frame))))))

Observe that applying frame-coord-map to a frame returns a procedure
that, given a vector, returns a vector. If the argument vector is in the unit
square, the result vector will be in the frame. For example,
((frame-coord-map a-frame) (make-vect 0 0))

returns the same vector as
(origin-frame a-frame)

      Exercise 2.46: A two-dimensional vector v running from
      the origin to a point can be represented as a pair consisting
      of an x-coordinate and a y-coordinate. Implement a data
      abstraction for vectors by giving a constructor make-vect
      and corresponding selectors xcor-vect and ycor-vect. In
      terms of your selectors and constructor, implement proce-
      dures add-vect, sub-vect, and scale-vect that perform
      the operations vector addition, vector subtraction, and mul-
      tiplying a vector by a scalar:
                  (x 1 , y1 ) + (x 2 , y2 ) = (x 1 + x 2 , y1 + y2 ),
                  (x 1 , y1 ) − (x 2 , y2 ) = (x 1 − x 2 , y1 − y2 ),
                               s · (x , y) = (sx , sy).
      Exercise 2.47: Here are two possible constructors for frames:
      (define (make-frame origin edge1 edge2)
         (list origin edge1 edge2))
      (define (make-frame origin edge1 edge2)
         (cons origin (cons edge1 edge2)))

      For each constructor supply the appropriate selectors to
      produce an implementation for frames.

A painter is represented as a procedure that, given a frame as argument,
draws a particular image shied and scaled to fit the frame. at is to
say, if p is a painter and f is a frame, then we produce p’s image in f by
calling p with f as argument.
     e details of how primitive painters are implemented depend on
the particular characteristics of the graphics system and the type of im-
age to be drawn. For instance, suppose we have a procedure draw-line
that draws a line on the screen between two specified points. en we
can create painters for line drawings, such as the wave painter in Figure
2.10, from lists of line segments as follows:27
(define (segments->painter segment-list)
  (lambda (frame)
      (lambda (segment)
          ((frame-coord-map frame)
            (start-segment segment))
          ((frame-coord-map frame)
            (end-segment segment))))

e segments are given using coordinates with respect to the unit square.
For each segment in the list, the painter transforms the segment end-
points with the frame coordinate map and draws a line between the
transformed points.
    Representing painters as procedures erects a powerful abstraction
barrier in the picture language. We can create and intermix all sorts of
  27 segments->painter     uses the representation for line segments described in Exer-
cise 2.48 below. It also uses the for-each procedure described in Exercise 2.23.

primitive painters, based on a variety of graphics capabilities. e de-
tails of their implementation do not maer. Any procedure can serve as
a painter, provided that it takes a frame as argument and draws some-
thing scaled to fit the frame.28
       Exercise 2.48: A directed line segment in the plane can be
       represented as a pair of vectors—the vector running from
       the origin to the start-point of the segment, and the vector
       running from the origin to the end-point of the segment.
       Use your vector representation from Exercise 2.46 to de-
       fine a representation for segments with a constructor make-
       segment and selectors start-segment and end-segment.

       Exercise 2.49: Use segments->painter to define the fol-
       lowing primitive painters:

           a. e painter that draws the outline of the designated
           b. e painter that draws an “X” by connecting opposite
              corners of the frame.
           c. e painter that draws a diamond shape by connect-
              ing the midpoints of the sides of the frame.
           d. e wave painter.
  28 For example, the rogers painter of Figure 2.11 was constructed from a gray-level
image. For each point in a given frame, the rogers painter determines the point in the
image that is mapped to it under the frame coordinate map, and shades it accordingly.
By allowing different types of painters, we are capitalizing on the abstract data idea
discussed in Section 2.1.3, where we argued that a rational-number representation could
be anything at all that satisfies an appropriate condition. Here we’re using the fact
that a painter can be implemented in any way at all, so long as it draws something
in the designated frame. Section 2.1.3 also showed how pairs could be implemented as
procedures. Painters are our second example of a procedural representation for data.

Transforming and combining painters
An operation on painters (such as flip-vert or beside) works by cre-
ating a painter that invokes the original painters with respect to frames
derived from the argument frame. us, for example, flip-vert doesn’t
have to know how a painter works in order to flip it—it just has to know
how to turn a frame upside down: e flipped painter just uses the orig-
inal painter, but in the inverted frame.
    Painter operations are based on the procedure transform-painter,
which takes as arguments a painter and information on how to trans-
form a frame and produces a new painter. e transformed painter,
when called on a frame, transforms the frame and calls the original
painter on the transformed frame. e arguments to transform-painter
are points (represented as vectors) that specify the corners of the new
frame: When mapped into the frame, the first point specifies the new
frame’s origin and the other two specify the ends of its edge vectors.
us, arguments within the unit square specify a frame contained within
the original frame.
(define (transform-painter painter origin corner1 corner2)
  (lambda (frame)
    (let ((m (frame-coord-map frame)))
      (let ((new-origin (m origin)))
        (painter (make-frame
                    (sub-vect (m corner1) new-origin)
                    (sub-vect (m corner2) new-origin)))))))

Here’s how to flip painter images vertically:
(define (flip-vert painter)
  (transform-painter painter
                       (make-vect 0.0 1.0)     ; new origin

                           (make-vect 1.0 1.0)           ; new end of edge1
                           (make-vect 0.0 0.0)))         ; new end of edge2
Using transform-painter, we can easily define new transformations.
For example, we can define a painter that shrinks its image to the upper-
right quarter of the frame it is given:
(define (shrink-to-upper-right painter)
   painter (make-vect 0.5 0.5)
   (make-vect 1.0 0.5) (make-vect 0.5 1.0)))

Other transformations rotate images counterclockwise by 90 degrees29
(define (rotate90 painter)
  (transform-painter painter
                           (make-vect 1.0 0.0)
                           (make-vect 1.0 1.0)
                           (make-vect 0.0 0.0)))

or squash images towards the center of the frame:30
(define (squash-inwards painter)
  (transform-painter painter
                           (make-vect 0.0 0.0)
                           (make-vect 0.65 0.35)
                           (make-vect 0.35 0.65)))

Frame transformation is also the key to defining means of combining
two or more painters. e beside procedure, for example, takes two
painters, transforms them to paint in the le and right halves of an
argument frame respectively, and produces a new, compound painter.
  29 rotate90 is a pure rotation only for square frames, because it also stretches and
shrinks the image to fit into the rotated frame.
  30 e diamond-shaped images in Figure 2.10 and Figure 2.11 were created with

squash-inwards applied to wave and rogers.

When the compound painter is given a frame, it calls the first trans-
formed painter to paint in the le half of the frame and calls the second
transformed painter to paint in the right half of the frame:
(define (beside painter1 painter2)
  (let ((split-point (make-vect 0.5 0.0)))
    (let ((paint-left
             (make-vect 0.0 0.0)
             (make-vect 0.0 1.0)))
             (make-vect 1.0 0.0)
             (make-vect 0.5 1.0))))
      (lambda (frame)
         (paint-left frame)
         (paint-right frame)))))

Observe how the painter data abstraction, and in particular the repre-
sentation of painters as procedures, makes beside easy to implement.
e beside procedure need not know anything about the details of the
component painters other than that each painter will draw something
in its designated frame.

      Exercise 2.50: Define the transformation flip-horiz, which
      flips painters horizontally, and transformations that rotate
      painters counterclockwise by 180 degrees and 270 degrees.

      Exercise 2.51: Define the below operation for painters. below
      takes two painters as arguments. e resulting painter, given

      a frame, draws with the first painter in the boom of the
      frame and with the second painter in the top. Define below
      in two different ways—first by writing a procedure that is
      analogous to the beside procedure given above, and again
      in terms of beside and suitable rotation operations (from
      Exercise 2.50).

Levels of language for robust design
e picture language exercises some of the critical ideas we’ve intro-
duced about abstraction with procedures and data. e fundamental
data abstractions, painters, are implemented using procedural represen-
tations, which enables the language to handle different basic drawing
capabilities in a uniform way. e means of combination satisfy the
closure property, which permits us to easily build up complex designs.
Finally, all the tools for abstracting procedures are available to us for
abstracting means of combination for painters.
     We have also obtained a glimpse of another crucial idea about lan-
guages and program design. is is the approach of stratified design,
the notion that a complex system should be structured as a sequence
of levels that are described using a sequence of languages. Each level is
constructed by combining parts that are regarded as primitive at that
level, and the parts constructed at each level are used as primitives at
the next level. e language used at each level of a stratified design has
primitives, means of combination, and means of abstraction appropriate
to that level of detail.
     Stratified design pervades the engineering of complex systems. For
example, in computer engineering, resistors and transistors are com-
bined (and described using a language of analog circuits) to produce
parts such as and-gates and or-gates, which form the primitives of a

language for digital-circuit design.31 ese parts are combined to build
processors, bus structures, and memory systems, which are in turn com-
bined to form computers, using languages appropriate to computer ar-
chitecture. Computers are combined to form distributed systems, using
languages appropriate for describing network interconnections, and so
    As a tiny example of stratification, our picture language uses prim-
itive elements (primitive painters) that are created using a language
that specifies points and lines to provide the lists of line segments for
segments->painter, or the shading details for a painter like rogers.
e bulk of our description of the picture language focused on com-
bining these primitives, using geometric combiners such as beside and
below. We also worked at a higher level, regarding beside and below
as primitives to be manipulated in a language whose operations, such
as square-of-four, capture common paerns of combining geometric
    Stratified design helps make programs robust, that is, it makes it
likely that small changes in a specification will require correspondingly
small changes in the program. For instance, suppose we wanted to change
the image based on wave shown in Figure 2.9. We could work at the
lowest level to change the detailed appearance of the wave element;
we could work at the middle level to change the way corner-split
replicates the wave; we could work at the highest level to change how
square-limit arranges the four copies of the corner. In general, each
level of a stratified design provides a different vocabulary for express-
ing the characteristics of the system, and a different kind of ability to
change it.

 31 Section   3.3.4 describes one such language.

      Exercise 2.52: Make changes to the square limit of wave
      shown in Figure 2.9 by working at each of the levels de-
      scribed above. In particular:

        a. Add some segments to the primitive wave painter of
           Exercise 2.49 (to add a smile, for example).
        b. Change the paern constructed by corner-split (for
           example, by using only one copy of the up-split and
           right-split images instead of two).

        c. Modify the version of square-limit that uses square-
           of-four so as to assemble the corners in a different
           paern. (For example, you might make the big Mr.
           Rogers look outward from each corner of the square.)

2.3 Symbolic Data
All the compound data objects we have used so far were constructed ul-
timately from numbers. In this section we extend the representational
capability of our language by introducing the ability to work with arbi-
trary symbols as data.

2.3.1 otation
If we can form compound data using symbols, we can have lists such as
(a b c d)
(23 45 17)
((Norah 12) (Molly 9) (Anna 7) (Lauren 6) (Charlotte 4))

Lists containing symbols can look just like the expressions of our lan-

(* (+ 23 45)
    (+ x 9))
(define (fact n)
  (if (= n 1) 1 (* n (fact (- n 1)))))

In order to manipulate symbols we need a new element in our language:
the ability to quote a data object. Suppose we want to construct the list
(a b). We can’t accomplish this with (list a b), because this expres-
sion constructs a list of the values of a and b rather than the symbols
themselves. is issue is well known in the context of natural languages,
where words and sentences may be regarded either as semantic entities
or as character strings (syntactic entities). e common practice in nat-
ural languages is to use quotation marks to indicate that a word or a
sentence is to be treated literally as a string of characters. For instance,
the first leer of “John” is clearly “J.” If we tell somebody “say your
name aloud,” we expect to hear that person’s name. However, if we tell
somebody “say ‘your name’ aloud,” we expect to hear the words “your
name.” Note that we are forced to nest quotation marks to describe what
somebody else might say.32
    We can follow this same practice to identify lists and symbols that
are to be treated as data objects rather than as expressions to be evalu-
  32 Allowing  quotation in a language wreaks havoc with the ability to reason about
the language in simple terms, because it destroys the notion that equals can be sub-
stituted for equals. For example, three is one plus two, but the word “three” is not the
phrase “one plus two.” otation is powerful because it gives us a way to build expres-
sions that manipulate other expressions (as we will see when we write an interpreter in
Chapter 4). But allowing statements in a language that talk about other statements in
that language makes it very difficult to maintain any coherent principle of what “equals
can be substituted for equals” should mean. For example, if we know that the evening
star is the morning star, then from the statement “the evening star is Venus” we can
deduce “the morning star is Venus.” However, given that “John knows that the evening
star is Venus” we cannot infer that “John knows that the morning star is Venus.”

ated. However, our format for quoting differs from that of natural lan-
guages in that we place a quotation mark (traditionally, the single quote
symbol ') only at the beginning of the object to be quoted. We can get
away with this in Scheme syntax because we rely on blanks and paren-
theses to delimit objects. us, the meaning of the single quote character
is to quote the next object.33
     Now we can distinguish between symbols and their values:
(define a 1)
(define b 2)
(list a b)
(1 2)
(list 'a 'b)
(a b)
(list 'a b)
(a 2)

otation also allows us to type in compound objects, using the con-
ventional printed representation for lists:34
  33 e    single quote is different from the double quote we have been using to enclose
character strings to be printed. Whereas the single quote can be used to denote lists or
symbols, the double quote is used only with character strings. In this book, the only
use for character strings is as items to be printed.
   34 Strictly, our use of the quotation mark violates the general rule that all compound

expressions in our language should be delimited by parentheses and look like lists. We
can recover this consistency by introducing a special form quote, which serves the
same purpose as the quotation mark. us, we would type (quote a) instead of 'a,
and we would type (quote (a b c)) instead of '(a b c). is is precisely how
the interpreter works. e quotation mark is just a single-character abbreviation for
wrapping the next complete expression with quote to form (quote ⟨ expression ⟩).
is is important because it maintains the principle that any expression seen by the
interpreter can be manipulated as a data object. For instance, we could construct the
expression (car '(a b c)), which is the same as (car (quote (a b c))), by evaluating
(list 'car (list 'quote '(a b c))).

(car '(a b c))
(cdr '(a b c))
(b c)

In keeping with this, we can obtain the empty list by evaluating '(),
and thus dispense with the variable nil.
     One additional primitive used in manipulating symbols is eq?, which
takes two symbols as arguments and tests whether they are the same.35
Using eq?, we can implement a useful procedure called memq. is takes
two arguments, a symbol and a list. If the symbol is not contained in the
list (i.e., is not eq? to any item in the list), then memq returns false. Other-
wise, it returns the sublist of the list beginning with the first occurrence
of the symbol:
(define (memq item x)
    (cond ((null? x) false)
            ((eq? item (car x)) x)
            (else (memq item (cdr x)))))

For example, the value of
(memq 'apple '(pear banana prune))

is false, whereas the value of
(memq 'apple '(x (apple sauce) y apple pear))

is (apple pear).

         Exercise 2.53: What would the interpreter print in response
         to evaluating each of the following expressions?
    35 We can consider two symbols to be “the same” if they consist of the same characters

in the same order. Such a definition skirts a deep issue that we are not yet ready to
address: the meaning of “sameness” in a programming language. We will return to this
in Chapter 3 (Section 3.1.3).

          (list 'a 'b 'c)
          (list (list 'george))
          (cdr '((x1 x2) (y1 y2)))
          (cadr '((x1 x2) (y1 y2)))
          (pair? (car '(a short list)))
          (memq 'red '((red shoes) (blue socks)))
          (memq 'red '(red shoes blue socks))

          Exercise 2.54: Two lists are said to be equal? if they con-
          tain equal elements arranged in the same order. For exam-
          (equal? '(this is a list) '(this is a list))

          is true, but
          (equal? '(this is a list) '(this (is a) list))

          is false. To be more precise, we can define equal? recur-
          sively in terms of the basic eq? equality of symbols by say-
          ing that a and b are equal? if they are both symbols and
          the symbols are eq?, or if they are both lists such that (car
          a) is equal? to (car b) and (cdr a) is equal? to (cdr b).
          Using this idea, implement equal? as a procedure.36

          Exercise 2.55: Eva Lu Ator types to the interpreter the ex-
  36 In practice, programmers use equal? to compare lists that contain numbers as
well as symbols. Numbers are not considered to be symbols. e question of whether
two numerically equal numbers (as tested by =) are also eq? is highly implementation-
dependent. A beer definition of equal? (such as the one that comes as a primitive in
Scheme) would also stipulate that if a and b are both numbers, then a and b are equal?
if they are numerically equal.

      (car ''abracadabra)

      To her surprise, the interpreter prints back quote. Explain.

2.3.2 Example: Symbolic Differentiation
As an illustration of symbol manipulation and a further illustration of
data abstraction, consider the design of a procedure that performs sym-
bolic differentiation of algebraic expressions. We would like the proce-
dure to take as arguments an algebraic expression and a variable and to
return the derivative of the expression with respect to the variable. For
example, if the arguments to the procedure are ax 2 + bx + c and x, the
procedure should return 2ax + b. Symbolic differentiation is of special
historical significance in Lisp. It was one of the motivating examples
behind the development of a computer language for symbol manipula-
tion. Furthermore, it marked the beginning of the line of research that
led to the development of powerful systems for symbolic mathematical
work, which are currently being used by a growing number of applied
mathematicians and physicists.
    In developing the symbolic-differentiation program, we will follow
the same strategy of data abstraction that we followed in developing
the rational-number system of Section 2.1.1. at is, we will first de-
fine a differentiation algorithm that operates on abstract objects such as
“sums,” “products,” and “variables” without worrying about how these
are to be represented. Only aerward will we address the representation

The differentiation program with abstract data
In order to keep things simple, we will consider a very simple symbolic-
differentiation program that handles expressions that are built up using

only the operations of addition and multiplication with two arguments.
Differentiation of any such expression can be carried out by applying
the following reduction rules:

          = 0,    for c a constant or a variable different from x ,
                                      = 1,
                           d (u + v ) du dv
                                      =    + ,
                               dx       dx dx
                           d (uv )     dv   du
                                   = u +v .
                             dx        dx   dx
Observe that the laer two rules are recursive in nature. at is, to ob-
tain the derivative of a sum we first find the derivatives of the terms and
add them. Each of the terms may in turn be an expression that needs
to be decomposed. Decomposing into smaller and smaller pieces will
eventually produce pieces that are either constants or variables, whose
derivatives will be either 0 or 1.
    To embody these rules in a procedure we indulge in a lile wishful
thinking, as we did in designing the rational-number implementation.
If we had a means for representing algebraic expressions, we should
be able to tell whether an expression is a sum, a product, a constant,
or a variable. We should be able to extract the parts of an expression.
For a sum, for example we want to be able to extract the addend (first
term) and the augend (second term). We should also be able to construct
expressions from parts. Let us assume that we already have procedures
to implement the following selectors, constructors, and predicates:
(variable? e)               Is e a variable?
(same-variable? v1 v2)      Are v1 and v2 the same variable?

(sum? e)                    Is e a sum?
(addend e)                  Addend of the sum e.
(augend e)                  Augend of the sum e.
(make-sum a1 a2)            Construct the sum of a1 and a2.
(product? e)                Is e a product?
(multiplier e)              Multiplier of the product e.
(multiplicand e)            Multiplicand of the product e.
(make-product m1 m2)        Construct the product of m1 and m2.

Using these, and the primitive predicate number?, which identifies num-
bers, we can express the differentiation rules as the following procedure:
(define (deriv exp var)
  (cond ((number? exp) 0)
         ((variable? exp) (if (same-variable? exp var) 1 0))
         ((sum? exp) (make-sum (deriv (addend exp) var)
                                 (deriv (augend exp) var)))
         ((product? exp)
             (make-product (multiplier exp)
                            (deriv (multiplicand exp) var))
             (make-product (deriv (multiplier exp) var)
                            (multiplicand exp))))
           (error "unknown expression type: DERIV" exp))))

is deriv procedure incorporates the complete differentiation algo-
rithm. Since it is expressed in terms of abstract data, it will work no
maer how we choose to represent algebraic expressions, as long as we
design a proper set of selectors and constructors. is is the issue we
must address next.

Representing algebraic expressions
We can imagine many ways to use list structure to represent algebraic
expressions. For example, we could use lists of symbols that mirror
the usual algebraic notation, representing ax + b as the list (a * x +
b). However, one especially straightforward choice is to use the same
parenthesized prefix notation that Lisp uses for combinations; that is,
to represent ax + b as (+ (* a x) b). en our data representation for
the differentiation problem is as follows:

    • e variables are symbols. ey are identified by the primitive
      predicate symbol?:
      (define (variable? x) (symbol? x))

    • Two variables are the same if the symbols representing them are

      (define (same-variable? v1 v2)
        (and (variable? v1) (variable? v2) (eq? v1 v2)))

    • Sums and products are constructed as lists:
      (define (make-sum a1 a2) (list '+ a1 a2))
      (define (make-product m1 m2) (list '* m1 m2))

    • A sum is a list whose first element is the symbol +:
      (define (sum? x) (and (pair? x) (eq? (car x) '+)))

    • e addend is the second item of the sum list:
      (define (addend s) (cadr s))

   • e augend is the third item of the sum list:
     (define (augend s) (caddr s))

   • A product is a list whose first element is the symbol *:
     (define (product? x) (and (pair? x) (eq? (car x) '*)))

   • e multiplier is the second item of the product list:
     (define (multiplier p) (cadr p))

   • e multiplicand is the third item of the product list:
     (define (multiplicand p) (caddr p))

us, we need only combine these with the algorithm as embodied by
deriv in order to have a working symbolic-differentiation program. Let
us look at some examples of its behavior:
(deriv '(+ x 3) 'x)
(+ 1 0)
(deriv '(* x y) 'x)
(+ (* x 0) (* 1 y))
(deriv '(* (* x y) (+ x 3)) 'x)
(+ (* (* x y) (+ 1 0))
   (* (+ (* x 0) (* 1 y))
      (+ x 3)))

e program produces answers that are correct; however, they are un-
simplified. It is true that

                         d (xy )
                                 = x · 0 + 1 · y,

but we would like the program to know that x · 0 = 0, 1 · y = y, and
0 + y = y. e answer for the second example should have been simply
y. As the third example shows, this becomes a serious issue when the
expressions are complex.
    Our difficulty is much like the one we encountered with the rational-
number implementation: we haven’t reduced answers to simplest form.
To accomplish the rational-number reduction, we needed to change only
the constructors and the selectors of the implementation. We can adopt
a similar strategy here. We won’t change deriv at all. Instead, we will
change make-sum so that if both summands are numbers, make-sum will
add them and return their sum. Also, if one of the summands is 0, then
make-sum will return the other summand.

(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
         ((=number? a2 0) a1)
         ((and (number? a1) (number? a2))
          (+ a1 a2))
         (else (list '+ a1 a2))))

is uses the procedure =number?, which checks whether an expression
is equal to a given number:
(define (=number? exp num) (and (number? exp) (= exp num)))

Similarly, we will change make-product to build in the rules that 0 times
anything is 0 and 1 times anything is the thing itself:
(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
         ((=number? m1 1) m2)
         ((=number? m2 1) m1)
         ((and (number? m1) (number? m2)) (* m1 m2))
         (else (list '* m1 m2))))

Here is how this version works on our three examples:
(deriv '(+ x 3) 'x)
(deriv '(* x y) 'x)
(deriv '(* (* x y) (+ x 3)) 'x)
(+ (* x y) (* y (+ x 3)))

Although this is quite an improvement, the third example shows that
there is still a long way to go before we get a program that puts ex-
pressions into a form that we might agree is “simplest.” e problem
of algebraic simplification is complex because, among other reasons, a
form that may be simplest for one purpose may not be for another.

     Exercise 2.56: Show how to extend the basic differentiator
     to handle more kinds of expressions. For instance, imple-
     ment the differentiation rule
                         d (un )          du
                                 = nun −1
                           dx             dx
     by adding a new clause to the deriv program and defining
     appropriate procedures exponentiation?, base, exponent,
     and make-exponentiation. (You may use the symbol **
     to denote exponentiation.) Build in the rules that anything
     raised to the power 0 is 1 and anything raised to the power
     1 is the thing itself.

     Exercise 2.57: Extend the differentiation program to han-
     dle sums and products of arbitrary numbers of (two or more)
     terms. en the last example above could be expressed as
     (deriv '(* x y (+ x 3)) 'x)

Try to do this by changing only the representation for sums
and products, without changing the deriv procedure at all.
For example, the addend of a sum would be the first term,
and the augend would be the sum of the rest of the terms.

Exercise 2.58: Suppose we want to modify the differentia-
tion program so that it works with ordinary mathematical
notation, in which + and * are infix rather than prefix opera-
tors. Since the differentiation program is defined in terms of
abstract data, we can modify it to work with different repre-
sentations of expressions solely by changing the predicates,
selectors, and constructors that define the representation of
the algebraic expressions on which the differentiator is to

  a. Show how to do this in order to differentiate algebraic
     expressions presented in infix form, such as (x + (3
     * (x + (y + 2)))). To simplify the task, assume that
     + and * always take two arguments and that expres-
     sions are fully parenthesized.
  b. e problem becomes substantially harder if we allow
     standard algebraic notation, such as (x + 3 * (x +
     y + 2)), which drops unnecessary parentheses and
     assumes that multiplication is done before addition.
     Can you design appropriate predicates, selectors, and
     constructors for this notation such that our derivative
     program still works?

2.3.3 Example: Representing Sets
In the previous examples we built representations for two kinds of com-
pound data objects: rational numbers and algebraic expressions. In one
of these examples we had the choice of simplifying (reducing) the ex-
pressions at either construction time or selection time, but other than
that the choice of a representation for these structures in terms of lists
was straightforward. When we turn to the representation of sets, the
choice of a representation is not so obvious. Indeed, there are a num-
ber of possible representations, and they differ significantly from one
another in several ways.
    Informally, a set is simply a collection of distinct objects. To give
a more precise definition we can employ the method of data abstrac-
tion. at is, we define “set” by specifying the operations that are to be
used on sets. ese are union-set, intersection-set, element-of-
set?, and adjoin-set. element-of-set? is a predicate that determines
whether a given element is a member of a set. adjoin-set takes an ob-
ject and a set as arguments and returns a set that contains the elements
of the original set and also the adjoined element. union-set computes
the union of two sets, which is the set containing each element that
appears in either argument. intersection-set computes the intersec-
tion of two sets, which is the set containing only elements that appear
in both arguments. From the viewpoint of data abstraction, we are free
to design any representation that implements these operations in a way
consistent with the interpretations given above.37
   37 If we want to be more formal, we can specify “consistent with the interpretations

given above” to mean that the operations satisfy a collection of rules such as these:
• For any set S and any object x, (element-of-set? x (adjoin-set x S)) is true
(informally: “Adjoining an object to a set produces a set that contains the object”).
• For any sets S and T and any object x, (element-of-set? x (union-set S T)) is

Sets as unordered lists
One way to represent a set is as a list of its elements in which no el-
ement appears more than once. e empty set is represented by the
empty list. In this representation, element-of-set? is similar to the
procedure memq of Section 2.3.1. It uses equal? instead of eq? so that
the set elements need not be symbols:
(define (element-of-set? x set)
  (cond ((null? set) false)
          ((equal? x (car set)) true)
          (else (element-of-set? x (cdr set)))))

Using this, we can write adjoin-set. If the object to be adjoined is al-
ready in the set, we just return the set. Otherwise, we use cons to add
the object to the list that represents the set:
(define (adjoin-set x set)
  (if (element-of-set? x set)
       (cons x set)))

For intersection-set we can use a recursive strategy. If we know how
to form the intersection of set2 and the cdr of set1, we only need to
decide whether to include the car of set1 in this. But this depends on
whether (car set1) is also in set2. Here is the resulting procedure:
(define (intersection-set set1 set2)
  (cond ((or (null? set1) (null? set2)) '())
          ((element-of-set? (car set1) set2)
           (cons (car set1) (intersection-set (cdr set1) set2)))
          (else (intersection-set (cdr set1) set2))))

equal to (or (element-of-set? x S) (element-of-set? x T)) (informally: “e
elements of (union S T) are the elements that are in S or in T”).
• For any object x, (element-of-set? x '()) is false (informally: “No object is an
element of the empty set”).
In designing a representation, one of the issues we should be concerned
with is efficiency. Consider the number of steps required by our set
operations. Since they all use element-of-set?, the speed of this oper-
ation has a major impact on the efficiency of the set implementation as
a whole. Now, in order to check whether an object is a member of a set,
element-of-set? may have to scan the entire set. (In the worst case,
the object turns out not to be in the set.) Hence, if the set has n elements,
element-of-set? might take up to n steps. us, the number of steps
required grows as Θ(n). e number of steps required by adjoin-set,
which uses this operation, also grows as Θ(n). For intersection-set,
which does an element-of-set? check for each element of set1, the
number of steps required grows as the product of the sizes of the sets
involved, or Θ(n 2 ) for two sets of size n. e same will be true of union-

      Exercise 2.59: Implement the union-set operation for the
      unordered-list representation of sets.

      Exercise 2.60: We specified that a set would be represented
      as a list with no duplicates. Now suppose we allow dupli-
      cates. For instance, the set {1, 2, 3} could be represented as
      the list (2 3 2 1 3 2 2). Design procedures element-
      of-set?, adjoin-set, union-set, and intersection-set
      that operate on this representation. How does the efficiency
      of each compare with the corresponding procedure for the
      non-duplicate representation? Are there applications for which
      you would use this representation in preference to the non-
      duplicate one?

Sets as ordered lists
One way to speed up our set operations is to change the representation
so that the set elements are listed in increasing order. To do this, we
need some way to compare two objects so that we can say which is
bigger. For example, we could compare symbols lexicographically, or
we could agree on some method for assigning a unique number to an
object and then compare the elements by comparing the corresponding
numbers. To keep our discussion simple, we will consider only the case
where the set elements are numbers, so that we can compare elements
using > and <. We will represent a set of numbers by listing its elements
in increasing order. Whereas our first representation above allowed us
to represent the set {1, 3, 6, 10} by listing the elements in any order, our
new representation allows only the list (1 3 6 10).
    One advantage of ordering shows up in element-of-set?: In check-
ing for the presence of an item, we no longer have to scan the entire set.
If we reach a set element that is larger than the item we are looking for,
then we know that the item is not in the set:
(define (element-of-set? x set)
  (cond ((null? set) false)
         ((= x (car set)) true)
         ((< x (car set)) false)
         (else (element-of-set? x (cdr set)))))

How many steps does this save? In the worst case, the item we are
looking for may be the largest one in the set, so the number of steps
is the same as for the unordered representation. On the other hand, if
we search for items of many different sizes we can expect that some-
times we will be able to stop searching at a point near the beginning of
the list and that other times we will still need to examine most of the
list. On the average we should expect to have to examine about half of

the items in the set. us, the average number of steps required will be
about n/2. is is still Θ(n) growth, but it does save us, on the average,
a factor of 2 in number of steps over the previous implementation.
     We obtain a more impressive speedup with intersection-set. In
the unordered representation this operation required Θ(n 2 ) steps, be-
cause we performed a complete scan of set2 for each element of set1.
But with the ordered representation, we can use a more clever method.
Begin by comparing the initial elements, x1 and x2, of the two sets. If
x1 equals x2, then that gives an element of the intersection, and the rest
of the intersection is the intersection of the cdr-s of the two sets. Sup-
pose, however, that x1 is less than x2. Since x2 is the smallest element
in set2, we can immediately conclude that x1 cannot appear anywhere
in set2 and hence is not in the intersection. Hence, the intersection is
equal to the intersection of set2 with the cdr of set1. Similarly, if x2
is less than x1, then the intersection is given by the intersection of set1
with the cdr of set2. Here is the procedure:
(define (intersection-set set1 set2)
  (if (or (null? set1) (null? set2))
      (let ((x1 (car set1)) (x2 (car set2)))
         (cond ((= x1 x2)
                 (cons x1 (intersection-set (cdr set1)
                                                (cdr set2))))
                ((< x1 x2)
                 (intersection-set (cdr set1) set2))
                ((< x2 x1)
                 (intersection-set set1 (cdr set2)))))))

To estimate the number of steps required by this process, observe that
at each step we reduce the intersection problem to computing inter-
sections of smaller sets—removing the first element from set1 or set2

or both. us, the number of steps required is at most the sum of the
sizes of set1 and set2, rather than the product of the sizes as with the
unordered representation. is is Θ(n) growth rather than Θ(n 2 )—a con-
siderable speedup, even for sets of moderate size.

      Exercise 2.61: Give an implementation of adjoin-set us-
      ing the ordered representation. By analogy with element-
      of-set? show how to take advantage of the ordering to
      produce a procedure that requires on the average about half
      as many steps as with the unordered representation.

      Exercise 2.62: Give a Θ(n) implementation of union-set
      for sets represented as ordered lists.

Sets as binary trees
We can do beer than the ordered-list representation by arranging the
set elements in the form of a tree. Each node of the tree holds one ele-
ment of the set, called the “entry” at that node, and a link to each of two
other (possibly empty) nodes. e “le” link points to elements smaller
than the one at the node, and the “right” link to elements greater than
the one at the node. Figure 2.16 shows some trees that represent the set
{1, 3, 5, 7, 9, 11}. e same set may be represented by a tree in a number
of different ways. e only thing we require for a valid representation
is that all elements in the le subtree be smaller than the node entry
and that all elements in the right subtree be larger.
     e advantage of the tree representation is this: Suppose we want to
check whether a number x is contained in a set. We begin by comparing
x with the entry in the top node. If x is less than this, we know that we
need only search the le subtree; if x is greater, we need only search
the right subtree. Now, if the tree is “balanced,” each of these subtrees

                        7                3                          5

                    3       9        1       7                  3       9

                1       5       11       5         9        1       7       11


       Figure 2.16: Various binary trees that represent the set
       {1, 3, 5, 7, 9, 11}.

will be about half the size of the original. us, in one step we have
reduced the problem of searching a tree of size n to searching a tree
of size n/2. Since the size of the tree is halved at each step, we should
expect that the number of steps needed to search a tree of size n grows
as Θ(log n).38 For large sets, this will be a significant speedup over the
previous representations.
    We can represent trees by using lists. Each node will be a list of three
items: the entry at the node, the le subtree, and the right subtree. A le
or a right subtree of the empty list will indicate that there is no subtree
connected there. We can describe this representation by the following
   38 Halving the size of the problem at each step is the distinguishing characteristic of

logarithmic growth, as we saw with the fast-exponentiation algorithm of Section 1.2.4
and the half-interval search method of Section 1.3.3.
   39 We are representing sets in terms of trees, and trees in terms of lists—in effect, a

data abstraction built upon a data abstraction. We can regard the procedures entry,
left-branch, right-branch, and make-tree as a way of isolating the abstraction of a
“binary tree” from the particular way we might wish to represent such a tree in terms
of list structure.

(define (entry tree) (car tree))
(define (left-branch tree) (cadr tree))
(define (right-branch tree) (caddr tree))
(define (make-tree entry left right)
  (list entry left right))

Now we can write the element-of-set? procedure using the strategy
described above:
(define (element-of-set? x set)
  (cond ((null? set) false)
         ((= x (entry set)) true)
         ((< x (entry set))
          (element-of-set? x (left-branch set)))
         ((> x (entry set))
          (element-of-set? x (right-branch set)))))

Adjoining an item to a set is implemented similarly and also requires
Θ(log n) steps. To adjoin an item x, we compare x with the node en-
try to determine whether x should be added to the right or to the le
branch, and having adjoined x to the appropriate branch we piece this
newly constructed branch together with the original entry and the other
branch. If x is equal to the entry, we just return the node. If we are asked
to adjoin x to an empty tree, we generate a tree that has x as the entry
and empty right and le branches. Here is the procedure:
(define (adjoin-set x set)
  (cond ((null? set) (make-tree x '() '()))
         ((= x (entry set)) set)
         ((< x (entry set))
          (make-tree (entry set)
                       (adjoin-set x (left-branch set))
                       (right-branch set)))
         ((> x (entry set))

           (make-tree (entry set) (left-branch set)
                          (adjoin-set x (right-branch set))))))

e above claim that searching the tree can be performed in a logarith-
mic number of steps rests on the assumption that the tree is “balanced,”
i.e., that the le and the right subtree of every tree have approximately
the same number of elements, so that each subtree contains about half
the elements of its parent. But how can we be certain that the trees we
construct will be balanced? Even if we start with a balanced tree, adding
elements with adjoin-set may produce an unbalanced result. Since the
position of a newly adjoined element depends on how the element com-
pares with the items already in the set, we can expect that if we add ele-
ments “randomly” the tree will tend to be balanced on the average. But
this is not a guarantee. For example, if we start with an empty set and
adjoin the numbers 1 through 7 in sequence we end up with the highly
unbalanced tree shown in Figure 2.17. In this tree all the le subtrees
are empty, so it has no advantage over a simple ordered list. One way to
solve this problem is to define an operation that transforms an arbitrary
tree into a balanced tree with the same elements. en we can perform
this transformation aer every few adjoin-set operations to keep our
set in balance. ere are also other ways to solve this problem, most of
which involve designing new data structures for which searching and
insertion both can be done in Θ(log n) steps.40

       Exercise 2.63: Each of the following two procedures con-
       verts a binary tree to a list.
       (define (tree->list-1 tree)
          (if (null? tree)

  40 Examples  of such structures include B-trees and red-black trees. ere is a large
literature on data structures devoted to this problem. See Cormen et al. 1990.


Figure 2.17: Unbalanced tree produced by adjoining 1
through 7 in sequence.

      (append (tree->list-1 (left-branch tree))
               (cons (entry tree)
                          (right-branch tree))))))
(define (tree->list-2 tree)
  (define (copy-to-list tree result-list)
    (if (null? tree)
        (copy-to-list (left-branch tree)
                          (cons (entry tree)
                                    (right-branch tree)
  (copy-to-list tree '()))

  a. Do the two procedures produce the same result for
     every tree? If not, how do the results differ? What lists

     do the two procedures produce for the trees in Figure
  b. Do the two procedures have the same order of growth
     in the number of steps required to convert a balanced
     tree with n elements to a list? If not, which one grows
     more slowly?

Exercise 2.64: e following procedure list->tree con-
verts an ordered list to a balanced binary tree. e helper
procedure partial-tree takes as arguments an integer n
and list of at least n elements and constructs a balanced
tree containing the first n elements of the list. e result re-
turned by partial-tree is a pair (formed with cons) whose
car is the constructed tree and whose cdr is the list of ele-
ments not included in the tree.
(define (list->tree elements)
  (car (partial-tree elements (length elements))))
(define (partial-tree elts n)
  (if (= n 0)
      (cons '() elts)
      (let ((left-size (quotient (- n 1) 2)))
         (let ((left-result
                 (partial-tree elts left-size)))
           (let ((left-tree (car left-result))
                  (non-left-elts (cdr left-result))
                  (right-size (- n (+ left-size 1))))
             (let ((this-entry (car non-left-elts))
                       (cdr non-left-elts)

                            (let ((right-tree (car right-result))
                                      (cdr right-result)))
                               (cons (make-tree this-entry

          a. Write a short paragraph explaining as clearly as you
             can how partial-tree works. Draw the tree produced
             by list->tree for the list (1 3 5 7 9 11).
          b. What is the order of growth in the number of steps re-
             quired by list->tree to convert a list of n elements?

       Exercise 2.65: Use the results of Exercise 2.63 and Exer-
       cise 2.64 to give Θ(n) implementations of union-set and
       intersection-set for sets implemented as (balanced) bi-
       nary trees.41

Sets and information retrieval
We have examined options for using lists to represent sets and have
seen how the choice of representation for a data object can have a large
impact on the performance of the programs that use the data. Another
reason for concentrating on sets is that the techniques discussed here
appear again and again in applications involving information retrieval.
    Consider a data base containing a large number of individual records,
such as the personnel files for a company or the transactions in an ac-
counting system. A typical data-management system spends a large
  41 Exercise   2.63 through Exercise 2.65 are due to Paul Hilfinger.

amount of time accessing or modifying the data in the records and
therefore requires an efficient method for accessing records. is is done
by identifying a part of each record to serve as an identifying key. A
key can be anything that uniquely identifies the record. For a personnel
file, it might be an employee’s  number. For an accounting system, it
might be a transaction number. Whatever the key is, when we define the
record as a data structure we should include a key selector procedure
that retrieves the key associated with a given record.
    Now we represent the data base as a set of records. To locate the
record with a given key we use a procedure lookup, which takes as
arguments a key and a data base and which returns the record that has
that key, or false if there is no such record. lookup is implemented in
almost the same way as element-of-set?. For example, if the set of
records is implemented as an unordered list, we could use
(define (lookup given-key set-of-records)
  (cond ((null? set-of-records) false)
        ((equal? given-key (key (car set-of-records)))
          (car set-of-records))
        (else (lookup given-key (cdr set-of-records)))))

Of course, there are beer ways to represent large sets than as un-
ordered lists. Information-retrieval systems in which records have to be
“randomly accessed” are typically implemented by a tree-based method,
such as the binary-tree representation discussed previously. In design-
ing such a system the methodology of data abstraction can be a great
help. e designer can create an initial implementation using a sim-
ple, straightforward representation such as unordered lists. is will be
unsuitable for the eventual system, but it can be useful in providing a
“quick and dirty” data base with which to test the rest of the system.
Later on, the data representation can be modified to be more sophisti-

cated. If the data base is accessed in terms of abstract selectors and con-
structors, this change in representation will not require any changes to
the rest of the system.

        Exercise 2.66: Implement the lookup procedure for the case
        where the set of records is structured as a binary tree, or-
        dered by the numerical values of the keys.

2.3.4 Example: Huffman Encoding Trees
is section provides practice in the use of list structure and data ab-
straction to manipulate sets and trees. e application is to methods for
representing data as sequences of ones and zeros (bits). For example,
the  standard code used to represent text in computers encodes
each character as a sequence of seven bits. Using seven bits allows us
to distinguish 27 , or 128, possible different characters. In general, if we
want to distinguish n different symbols, we will need to use log2 n bits
per symbol. If all our messages are made up of the eight symbols A, B,
C, D, E, F, G, and H, we can choose a code with three bits per character,
for example

A 000      C 010     E 100     G 110
B 001      D 011     F 101     H 111

With this code, the message


is encoded as the string of 54 bits


Codes such as  and the A-through-H code above are known as
fixed-length codes, because they represent each symbol in the message
with the same number of bits. It is sometimes advantageous to use variable-
length codes, in which different symbols may be represented by differ-
ent numbers of bits. For example, Morse code does not use the same
number of dots and dashes for each leer of the alphabet. In particular,
E, the most frequent leer, is represented by a single dot. In general, if
our messages are such that some symbols appear very frequently and
some very rarely, we can encode data more efficiently (i.e., using fewer
bits per message) if we assign shorter codes to the frequent symbols.
Consider the following alternative code for the leers A through H:

A 0       C 1010     E 1100     G 1110
B 100     D 1011     F 1101     H 1111

With this code, the same message as above is encoded as the string


is string contains 42 bits, so it saves more than 20% in space in com-
parison with the fixed-length code shown above.
    One of the difficulties of using a variable-length code is knowing
when you have reached the end of a symbol in reading a sequence of
zeros and ones. Morse code solves this problem by using a special sep-
arator code (in this case, a pause) aer the sequence of dots and dashes
for each leer. Another solution is to design the code in such a way that
no complete code for any symbol is the beginning (or prefix ) of the code
for another symbol. Such a code is called a prefix code. In the example
above, A is encoded by 0 and B is encoded by 100, so no other symbol
can have a code that begins with 0 or with 100.

     In general, we can aain significant savings if we use variable-length
prefix codes that take advantage of the relative frequencies of the sym-
bols in the messages to be encoded. One particular scheme for doing
this is called the Huffman encoding method, aer its discoverer, David
Huffman. A Huffman code can be represented as a binary tree whose
leaves are the symbols that are encoded. At each non-leaf node of the
tree there is a set containing all the symbols in the leaves that lie below
the node. In addition, each symbol at a leaf is assigned a weight (which
is its relative frequency), and each non-leaf node contains a weight that
is the sum of all the weights of the leaves lying below it. e weights
are not used in the encoding or the decoding process. We will see below
how they are used to help construct the tree.
     Figure 2.18 shows the Huffman tree for the A-through-H code given
above. e weights at the leaves indicate that the tree was designed for
messages in which A appears with relative frequency 8, B with relative
frequency 3, and the other leers each with relative frequency 1.
     Given a Huffman tree, we can find the encoding of any symbol by
starting at the root and moving down until we reach the leaf that holds
the symbol. Each time we move down a le branch we add a 0 to the
code, and each time we move down a right branch we add a 1. (We
decide which branch to follow by testing to see which branch either
is the leaf node for the symbol or contains the symbol in its set.) For
example, starting from the root of the tree in Figure 2.18, we arrive at
the leaf for D by following a right branch, then a le branch, then a right
branch, then a right branch; hence, the code for D is 1011.
     To decode a bit sequence using a Huffman tree, we begin at the root
and use the successive zeros and ones of the bit sequence to determine
whether to move down the le or the right branch. Each time we come
to a leaf, we have generated a new symbol in the message, at which

                    {A B C D E F G H} 17

                                     {B C D E F G H} 9

                         A 8

                                               {E F G H} 4

          {B C D} 5              {E F} 2

                           {C D} 2                       {G H} 2

            B 3                  E 1          F 1

                   C 1         D 1               G 1          H 1

                  Figure 2.18: A Huffman encoding tree.

point we start over from the root of the tree to find the next symbol.
For example, suppose we are given the tree above and the sequence
10001010. Starting at the root, we move down the right branch, (since
the first bit of the string is 1), then down the le branch (since the second
bit is 0), then down the le branch (since the third bit is also 0). is
brings us to the leaf for B, so the first symbol of the decoded message is
B. Now we start again at the root, and we make a le move because the
next bit in the string is 0. is brings us to the leaf for A. en we start
again at the root with the rest of the string 1010, so we move right, le,
right, le and reach C. us, the entire message is BAC.

Generating Huffman trees
Given an “alphabet” of symbols and their relative frequencies, how do
we construct the “best” code? (In other words, which tree will encode
messages with the fewest bits?) Huffman gave an algorithm for doing

this and showed that the resulting code is indeed the best variable-
length code for messages where the relative frequency of the symbols
matches the frequencies with which the code was constructed. We will
not prove this optimality of Huffman codes here, but we will show how
Huffman trees are constructed.42
    e algorithm for generating a Huffman tree is very simple. e idea
is to arrange the tree so that the symbols with the lowest frequency
appear farthest away from the root. Begin with the set of leaf nodes,
containing symbols and their frequencies, as determined by the initial
data from which the code is to be constructed. Now find two leaves with
the lowest weights and merge them to produce a node that has these
two nodes as its le and right branches. e weight of the new node is
the sum of the two weights. Remove the two leaves from the original
set and replace them by this new node. Now continue this process. At
each step, merge two nodes with the smallest weights, removing them
from the set and replacing them with a node that has these two as its
le and right branches. e process stops when there is only one node
le, which is the root of the entire tree. Here is how the Huffman tree
of Figure 2.18 was generated:

Initial leaves        {(A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (G 1) (H 1)}
             Merge    {(A 8) (B 3) ({C D} 2) (E 1) (F 1) (G 1) (H 1)}
             Merge    {(A 8) (B 3) ({C D} 2) ({E F} 2) (G 1) (H 1)}
             Merge    {(A 8) (B 3) ({C D} 2) ({E F} 2) ({G H} 2)}
             Merge    {(A 8) (B 3) ({C D} 2) ({E F G H} 4)}
             Merge    {(A 8) ({B C D} 5) ({E F G H} 4)}
             Merge    {(A 8) ({B C D E F G H} 9)}
   Final merge        {({A B C D E F G H} 17)}
  42 See   Hamming 1980 for a discussion of the mathematical properties of Huffman

e algorithm does not always specify a unique tree, because there may
not be unique smallest-weight nodes at each step. Also, the choice of the
order in which the two nodes are merged (i.e., which will be the right
branch and which will be the le branch) is arbitrary.

Representing Huffman trees
In the exercises below we will work with a system that uses Huffman
trees to encode and decode messages and generates Huffman trees ac-
cording to the algorithm outlined above. We will begin by discussing
how trees are represented.
    Leaves of the tree are represented by a list consisting of the symbol
leaf, the symbol at the leaf, and the weight:

(define (make-leaf symbol weight) (list 'leaf symbol weight))
(define (leaf? object) (eq? (car object) 'leaf))
(define (symbol-leaf x) (cadr x))
(define (weight-leaf x) (caddr x))

A general tree will be a list of a le branch, a right branch, a set of
symbols, and a weight. e set of symbols will be simply a list of the
symbols, rather than some more sophisticated set representation. When
we make a tree by merging two nodes, we obtain the weight of the
tree as the sum of the weights of the nodes, and the set of symbols as
the union of the sets of symbols for the nodes. Since our symbol sets
are represented as lists, we can form the union by using the append
procedure we defined in Section 2.2.1:
(define (make-code-tree left right)
  (list left
         (append (symbols left) (symbols right))
         (+ (weight left) (weight right))))

If we make a tree in this way, we have the following selectors:
(define (left-branch    tree) (car      tree))
(define (right-branch tree) (cadr tree))
(define (symbols tree)
  (if (leaf? tree)
      (list (symbol-leaf tree))
      (caddr tree)))
(define (weight tree)
  (if (leaf? tree)
      (weight-leaf tree)
      (cadddr tree)))

e procedures symbols and weight must do something slightly differ-
ent depending on whether they are called with a leaf or a general tree.
ese are simple examples of generic procedures (procedures that can
handle more than one kind of data), which we will have much more to
say about in Section 2.4 and Section 2.5.

The decoding procedure
e following procedure implements the decoding algorithm. It takes
as arguments a list of zeros and ones, together with a Huffman tree.
(define (decode bits tree)
  (define (decode-1 bits current-branch)
    (if (null? bits)
        (let ((next-branch
                (choose-branch (car bits) current-branch)))
           (if (leaf? next-branch)
               (cons (symbol-leaf next-branch)
                      (decode-1 (cdr bits) tree))
               (decode-1 (cdr bits) next-branch)))))
  (decode-1 bits tree))

(define (choose-branch bit branch)
  (cond ((= bit 0) (left-branch branch))
         ((= bit 1) (right-branch branch))
         (else (error "bad bit: CHOOSE-BRANCH" bit))))

e procedure decode-1 takes two arguments: the list of remaining bits
and the current position in the tree. It keeps moving “down” the tree,
choosing a le or a right branch according to whether the next bit in the
list is a zero or a one. (is is done with the procedure choose-branch.)
When it reaches a leaf, it returns the symbol at that leaf as the next
symbol in the message by consing it onto the result of decoding the
rest of the message, starting at the root of the tree. Note the error check
in the final clause of choose-branch, which complains if the procedure
finds something other than a zero or a one in the input data.

Sets of weighted elements
In our representation of trees, each non-leaf node contains a set of sym-
bols, which we have represented as a simple list. However, the tree-
generating algorithm discussed above requires that we also work with
sets of leaves and trees, successively merging the two smallest items.
Since we will be required to repeatedly find the smallest item in a set, it
is convenient to use an ordered representation for this kind of set.
    We will represent a set of leaves and trees as a list of elements, ar-
ranged in increasing order of weight. e following adjoin-set pro-
cedure for constructing sets is similar to the one described in Exercise
2.61; however, items are compared by their weights, and the element
being added to the set is never already in it.
(define (adjoin-set x set)
  (cond ((null? set) (list x))
         ((< (weight x) (weight (car set))) (cons x set))

        (else (cons (car set)
                      (adjoin-set x (cdr set))))))

e following procedure takes a list of symbol-frequency pairs such as
((A 4) (B 2) (C 1) (D 1)) and constructs an initial ordered set of
leaves, ready to be merged according to the Huffman algorithm:
(define (make-leaf-set pairs)
  (if (null? pairs)
      (let ((pair (car pairs)))
        (adjoin-set (make-leaf (car pair)       ; symbol
                                 (cadr pair))   ; frequency
                      (make-leaf-set (cdr pairs))))))

     Exercise 2.67: Define an encoding tree and a sample mes-
     (define sample-tree
        (make-code-tree (make-leaf 'A 4)
                          (make-leaf 'B 2)
                           (make-leaf 'D 1)
                           (make-leaf 'C 1)))))
     (define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))

     Use the decode procedure to decode the message, and give
     the result.

     Exercise 2.68: e encode procedure takes as arguments a
     message and a tree and produces the list of bits that gives
     the encoded message.

(define (encode message tree)
  (if (null? message)
       (append (encode-symbol (car message) tree)
                 (encode (cdr message) tree))))

encode-symbol     is a procedure, which you must write, that
returns the list of bits that encodes a given symbol accord-
ing to a given tree. You should design encode-symbol so
that it signals an error if the symbol is not in the tree at all.
Test your procedure by encoding the result you obtained in
Exercise 2.67 with the sample tree and seeing whether it is
the same as the original sample message.

Exercise 2.69: e following procedure takes as its argu-
ment a list of symbol-frequency pairs (where no symbol
appears in more than one pair) and generates a Huffman
encoding tree according to the Huffman algorithm.
(define (generate-huffman-tree pairs)
  (successive-merge (make-leaf-set pairs)))

make-leaf-set      is the procedure given above that trans-
forms the list of pairs into an ordered set of leaves. successive-
merge is the procedure you must write, using make-code-
tree to successively merge the smallest-weight elements
of the set until there is only one element le, which is the
desired Huffman tree. (is procedure is slightly tricky, but
not really complicated. If you find yourself designing a com-
plex procedure, then you are almost certainly doing some-
thing wrong. You can take significant advantage of the fact
that we are using an ordered set representation.)

Exercise 2.70: e following eight-symbol alphabet with
associated relative frequencies was designed to efficiently
encode the lyrics of 1950s rock songs. (Note that the “sym-
bols” of an “alphabet” need not be individual leers.)

A     2     GET 2    SHA 3     WAH 1
BOOM 1      JOB 2    NA 16     YIP 9

Use generate-huffman-tree (Exercise 2.69) to generate a
corresponding Huffman tree, and use encode (Exercise 2.68)
to encode the following message:

Get a job
Sha na na na na na na na na
Get a job
Sha na na na na na na na na
Wah yip yip yip yip yip yip yip yip yip
Sha boom

How many bits are required for the encoding? What is the
smallest number of bits that would be needed to encode this
song if we used a fixed-length code for the eight-symbol

Exercise 2.71: Suppose we have a Huffman tree for an al-
phabet of n symbols, and that the relative frequencies of
the symbols are 1, 2, 4, . . . , 2n −1 . Sketch the tree for n = 5;
for n = 10. In such a tree (for general n) how many bits
are required to encode the most frequent symbol? e least
frequent symbol?

      Exercise 2.72: Consider the encoding procedure that you
      designed in Exercise 2.68. What is the order of growth in
      the number of steps needed to encode a symbol? Be sure
      to include the number of steps needed to search the sym-
      bol list at each node encountered. To answer this question
      in general is difficult. Consider the special case where the
      relative frequencies of the n symbols are as described in Ex-
      ercise 2.71, and give the order of growth (as a function of n)
      of the number of steps needed to encode the most frequent
      and least frequent symbols in the alphabet.

2.4 Multiple Representations for Abstract Data
We have introduced data abstraction, a methodology for structuring
systems in such a way that much of a program can be specified indepen-
dent of the choices involved in implementing the data objects that the
program manipulates. For example, we saw in Section 2.1.1 how to sep-
arate the task of designing a program that uses rational numbers from
the task of implementing rational numbers in terms of the computer
language’s primitive mechanisms for constructing compound data. e
key idea was to erect an abstraction barrier—in this case, the selec-
tors and constructors for rational numbers (make-rat, numer, denom)—
that isolates the way rational numbers are used from their underlying
representation in terms of list structure. A similar abstraction barrier
isolates the details of the procedures that perform rational arithmetic
(add-rat, sub-rat, mul-rat, and div-rat) from the “higher-level” pro-
cedures that use rational numbers. e resulting program has the struc-
ture shown in Figure 2.1.
    ese data-abstraction barriers are powerful tools for controlling

complexity. By isolating the underlying representations of data objects,
we can divide the task of designing a large program into smaller tasks
that can be performed separately. But this kind of data abstraction is not
yet powerful enough, because it may not always make sense to speak
of “the underlying representation” for a data object.
    For one thing, there might be more than one useful representation
for a data object, and we might like to design systems that can deal with
multiple representations. To take a simple example, complex numbers
may be represented in two almost equivalent ways: in rectangular form
(real and imaginary parts) and in polar form (magnitude and angle).
Sometimes rectangular form is more appropriate and sometimes polar
form is more appropriate. Indeed, it is perfectly plausible to imagine a
system in which complex numbers are represented in both ways, and
in which the procedures for manipulating complex numbers work with
either representation.
    More importantly, programming systems are oen designed by many
people working over extended periods of time, subject to requirements
that change over time. In such an environment, it is simply not possi-
ble for everyone to agree in advance on choices of data representation.
So in addition to the data-abstraction barriers that isolate representa-
tion from use, we need abstraction barriers that isolate different de-
sign choices from each other and permit different choices to coexist in
a single program. Furthermore, since large programs are oen created
by combining pre-existing modules that were designed in isolation, we
need conventions that permit programmers to incorporate modules into
larger systems additively, that is, without having to redesign or reimple-
ment these modules.
    In this section, we will learn how to cope with data that may be
represented in different ways by different parts of a program. is re-

quires constructing generic procedures—procedures that can operate on
data that may be represented in more than one way. Our main technique
for building generic procedures will be to work in terms of data objects
that have type tags, that is, data objects that include explicit information
about how they are to be processed. We will also discuss data-directed
programming, a powerful and convenient implementation strategy for
additively assembling systems with generic operations.
    We begin with the simple complex-number example. We will see
how type tags and data-directed style enable us to design separate rect-
angular and polar representations for complex numbers while main-
taining the notion of an abstract “complex-number” data object. We will
accomplish this by defining arithmetic procedures for complex numbers
(add-complex, sub-complex, mul-complex, and div-complex) in terms
of generic selectors that access parts of a complex number independent
of how the number is represented. e resulting complex-number sys-
tem, as shown in Figure 2.19, contains two different kinds of abstrac-
tion barriers. e “horizontal” abstraction barriers play the same role
as the ones in Figure 2.1. ey isolate “higher-level” operations from
“lower-level” representations. In addition, there is a “vertical” barrier
that gives us the ability to separately design and install alternative rep-
    In Section 2.5 we will show how to use type tags and data-directed
style to develop a generic arithmetic package. is provides procedures
(add, mul, and so on) that can be used to manipulate all sorts of “num-
bers” and can be easily extended when a new kind of number is needed.
In Section 2.5.3, we’ll show how to use generic arithmetic in a system
that performs symbolic algebra.

                        Programs that use complex numbers

          add-complex       sub-complex      mul-complex       div-complex

                            Complex-arithmetic package

                   Rectangular                             Polar
                  representation                      representation

                  List structure and primitive machine arithmetic

       Figure 2.19: Data-abstraction barriers in the complex-
       number system.

2.4.1 Representations for Complex Numbers
We will develop a system that performs arithmetic operations on com-
plex numbers as a simple but unrealistic example of a program that uses
generic operations. We begin by discussing two plausible representa-
tions for complex numbers as ordered pairs: rectangular form (real part
and imaginary part) and polar form (magnitude and angle).43 Section
2.4.2 will show how both representations can be made to coexist in a
single system through the use of type tags and generic operations.
    Like rational numbers, complex numbers are naturally represented
as ordered pairs. e set of complex numbers can be thought of as a
two-dimensional space with two orthogonal axes, the “real” axis and the
  43 In actual computational systems, rectangular form is preferable to polar form most

of the time because of roundoff errors in conversion between rectangular and polar
form. is is why the complex-number example is unrealistic. Nevertheless, it provides
a clear illustration of the design of a system using generic operations and a good intro-
duction to the more substantial systems to be developed later in this chapter.


             y                                    z = x + iy = re iA



        Figure 2.20: Complex numbers as points in the plane.

“imaginary” axis. (See Figure 2.20.) From this point of view, the complex
number z = x + iy (where i 2 = −1) can be thought of as the point in the
plane whose real coordinate is x and whose imaginary coordinate is y.
Addition of complex numbers reduces in this representation to addition
of coordinates:
      Real-part(z 1 + z 2 ) = Real-part(z 1 ) + Real-part(z 2 ),
 Imaginary-part(z 1 + z 2 ) = Imaginary-part(z 1 ) + Imaginary-part(z 2 ).
    When multiplying complex numbers, it is more natural to think in
terms of representing a complex number in polar form, as a magnitude
and an angle (r and A in Figure 2.20). e product of two complex num-
bers is the vector obtained by stretching one complex number by the
length of the other and then rotating it through the angle of the other:

        Magnitude(z 1 · z 2 ) = Magnitude(z 1 ) · Magnitude(z 2 ),
           Angle(z 1 · z 2 ) = Angle(z 1 ) + Angle(z 2 ).
us, there are two different representations for complex numbers, which

are appropriate for different operations. Yet, from the viewpoint of some-
one writing a program that uses complex numbers, the principle of data
abstraction suggests that all the operations for manipulating complex
numbers should be available regardless of which representation is used
by the computer. For example, it is oen useful to be able to find the
magnitude of a complex number that is specified by rectangular coor-
dinates. Similarly, it is oen useful to be able to determine the real part
of a complex number that is specified by polar coordinates.
    To design such a system, we can follow the same data-abstraction
strategy we followed in designing the rational-number package in Sec-
tion 2.1.1. Assume that the operations on complex numbers are imple-
mented in terms of four selectors: real-part, imag-part, magnitude
and angle. Also assume that we have two procedures for construct-
ing complex numbers: make-from-real-imag returns a complex num-
ber with specified real and imaginary parts, and make-from-mag-ang
returns a complex number with specified magnitude and angle. ese
procedures have the property that, for any complex number z, both
(make-from-real-imag (real-part z) (imag-part z))

(make-from-mag-ang (magnitude z) (angle z))

produce complex numbers that are equal to z.
    Using these constructors and selectors, we can implement arith-
metic on complex numbers using the “abstract data” specified by the
constructors and selectors, just as we did for rational numbers in Sec-
tion 2.1.1. As shown in the formulas above, we can add and subtract
complex numbers in terms of real and imaginary parts while multiply-
ing and dividing complex numbers in terms of magnitudes and angles:

(define (add-complex z1 z2)
  (make-from-real-imag (+ (real-part z1) (real-part z2))
                          (+ (imag-part z1) (imag-part z2))))
(define (sub-complex z1 z2)
  (make-from-real-imag (- (real-part z1) (real-part z2))
                          (- (imag-part z1) (imag-part z2))))
(define (mul-complex z1 z2)
  (make-from-mag-ang (* (magnitude z1) (magnitude z2))
                        (+ (angle z1) (angle z2))))
(define (div-complex z1 z2)
  (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
                        (- (angle z1) (angle z2))))

To complete the complex-number package, we must choose a represen-
tation and we must implement the constructors and selectors in terms
of primitive numbers and primitive list structure. ere are two obvi-
ous ways to do this: We can represent a complex number in “rectangular
form” as a pair (real part, imaginary part) or in “polar form” as a pair
(magnitude, angle). Which shall we choose?
    In order to make the different choices concrete, imagine that there
are two programmers, Ben Bitdiddle and Alyssa P. Hacker, who are
independently designing representations for the complex-number sys-
tem. Ben chooses to represent complex numbers in rectangular form.
With this choice, selecting the real and imaginary parts of a complex
number is straightforward, as is constructing a complex number with
given real and imaginary parts. To find the magnitude and the angle, or
to construct a complex number with a given magnitude and angle, he
uses the trigonometric relations
                  x = r cos A,        r = x 2 + y2 ,
                  y = r sin A,       A = arctan(y, x),
which relate the real and imaginary parts (x , y) to the magnitude and the

angle (r, A).44 Ben’s representation is therefore given by the following
selectors and constructors:
(define (real-part z) (car z))
(define (imag-part z) (cdr z))
(define (magnitude z)
  (sqrt (+ (square (real-part z))
              (square (imag-part z)))))
(define (angle z)
  (atan (imag-part z) (real-part z)))
(define (make-from-real-imag x y) (cons x y))
(define (make-from-mag-ang r a)
  (cons (* r (cos a)) (* r (sin a))))

Alyssa, in contrast, chooses to represent complex numbers in polar form.
For her, selecting the magnitude and angle is straightforward, but she
has to use the trigonometric relations to obtain the real and imaginary
parts. Alyssa’s representation is:
(define (real-part z) (* (magnitude z) (cos (angle z))))
(define (imag-part z) (* (magnitude z) (sin (angle z))))
(define (magnitude z) (car z))
(define (angle z) (cdr z))
(define (make-from-real-imag x y)
  (cons (sqrt (+ (square x) (square y)))
          (atan y x)))
(define (make-from-mag-ang r a) (cons r a))

e discipline of data abstraction ensures that the same implementation
of add-complex, sub-complex, mul-complex, and div-complex will work
with either Ben’s representation or Alyssa’s representation.
  44 e   arctangent function referred to here, computed by Scheme’s atan procedure,
is defined so as to take two arguments y and x and to return the angle whose tangent
is y/x . e signs of the arguments determine the quadrant of the angle.

2.4.2 Tagged data
One way to view data abstraction is as an application of the “princi-
ple of least commitment.” In implementing the complex-number system
in Section 2.4.1, we can use either Ben’s rectangular representation or
Alyssa’s polar representation. e abstraction barrier formed by the se-
lectors and constructors permits us to defer to the last possible moment
the choice of a concrete representation for our data objects and thus
retain maximum flexibility in our system design.
    e principle of least commitment can be carried to even further
extremes. If we desire, we can maintain the ambiguity of representation
even aer we have designed the selectors and constructors, and elect
to use both Ben’s representation and Alyssa’s representation. If both
representations are included in a single system, however, we will need
some way to distinguish data in polar form from data in rectangular
form. Otherwise, if we were asked, for instance, to find the magnitude
of the pair (3, 4), we wouldn’t know whether to answer 5 (interpreting
the number in rectangular form) or 3 (interpreting the number in polar
form). A straightforward way to accomplish this distinction is to include
a type tag —the symbol rectangular or polar—as part of each complex
number. en when we need to manipulate a complex number we can
use the tag to decide which selector to apply.
    In order to manipulate tagged data, we will assume that we have
procedures type-tag and contents that extract from a data object the
tag and the actual contents (the polar or rectangular coordinates, in the
case of a complex number). We will also postulate a procedure attach-
tag that takes a tag and contents and produces a tagged data object. A
straightforward way to implement this is to use ordinary list structure:
(define (attach-tag type-tag contents)
  (cons type-tag contents))

(define (type-tag datum)
  (if (pair? datum)
      (car datum)
      (error "Bad tagged datum: TYPE-TAG" datum)))
(define (contents datum)
  (if (pair? datum)
      (cdr datum)
      (error "Bad tagged datum: CONTENTS" datum)))

Using these procedures, we can define predicates rectangular? and
polar?, which recognize rectangular and polar numbers, respectively:

(define (rectangular? z)
  (eq? (type-tag z) 'rectangular))
(define (polar? z) (eq? (type-tag z) 'polar))

With type tags, Ben and Alyssa can now modify their code so that their
two different representations can coexist in the same system. When-
ever Ben constructs a complex number, he tags it as rectangular. When-
ever Alyssa constructs a complex number, she tags it as polar. In addi-
tion, Ben and Alyssa must make sure that the names of their proce-
dures do not conflict. One way to do this is for Ben to append the suffix
rectangular to the name of each of his representation procedures and
for Alyssa to append polar to the names of hers. Here is Ben’s revised
rectangular representation from Section 2.4.1:
(define (real-part-rectangular z) (car z))
(define (imag-part-rectangular z) (cdr z))
(define (magnitude-rectangular z)
  (sqrt (+ (square (real-part-rectangular z))
            (square (imag-part-rectangular z)))))
(define (angle-rectangular z)
  (atan (imag-part-rectangular z)
        (real-part-rectangular z)))

(define (make-from-real-imag-rectangular x y)
  (attach-tag 'rectangular (cons x y)))
(define (make-from-mag-ang-rectangular r a)
  (attach-tag 'rectangular
               (cons (* r (cos a)) (* r (sin a)))))

and here is Alyssa’s revised polar representation:
(define (real-part-polar z)
  (* (magnitude-polar z) (cos (angle-polar z))))
(define (imag-part-polar z)
  (* (magnitude-polar z) (sin (angle-polar z))))
(define (magnitude-polar z) (car z))
(define (angle-polar z) (cdr z))
(define (make-from-real-imag-polar x y)
  (attach-tag 'polar
               (cons (sqrt (+ (square x) (square y)))
                      (atan y x))))
(define (make-from-mag-ang-polar r a)
  (attach-tag 'polar (cons r a)))

Each generic selector is implemented as a procedure that checks the tag
of its argument and calls the appropriate procedure for handling data
of that type. For example, to obtain the real part of a complex number,
real-part examines the tag to determine whether to use Ben’s real-
part-rectangular or Alyssa’s real-part-polar. In either case, we use
contents to extract the bare, untagged datum and send this to the rect-
angular or polar procedure as required:
(define (real-part z)
  (cond ((rectangular? z)
          (real-part-rectangular (contents z)))
        ((polar? z)
          (real-part-polar (contents z)))
        (else (error "Unknown type: REAL-PART" z))))

(define (imag-part z)
  (cond ((rectangular? z)
         (imag-part-rectangular (contents z)))
        ((polar? z)
         (imag-part-polar (contents z)))
        (else (error "Unknown type: IMAG-PART" z))))
(define (magnitude z)
  (cond ((rectangular? z)
         (magnitude-rectangular (contents z)))
        ((polar? z)
         (magnitude-polar (contents z)))
        (else (error "Unknown type: MAGNITUDE" z))))
(define (angle z)
  (cond ((rectangular? z)
         (angle-rectangular (contents z)))
        ((polar? z)
         (angle-polar (contents z)))
        (else (error "Unknown type: ANGLE" z))))

To implement the complex-number arithmetic operations, we can use
the same procedures add-complex, sub-complex, mul-complex, and div-
complex from Section 2.4.1, because the selectors they call are generic,
and so will work with either representation. For example, the procedure
add-complex is still
(define (add-complex z1 z2)
  (make-from-real-imag (+ (real-part z1) (real-part z2))
                         (+ (imag-part z1) (imag-part z2))))

Finally, we must choose whether to construct complex numbers using
Ben’s representation or Alyssa’s representation. One reasonable choice
is to construct rectangular numbers whenever we have real and imag-
inary parts and to construct polar numbers whenever we have magni-
tudes and angles:

                       Programs that use complex numbers

        add-complex       sub-complex    mul-complex   div-complex

                          Complex-arithmetic package

                           real-part     magnitude
                           imag-part         angle
       Rectangular                                              Polar
      representation                                       representation

               List structure and primitive machine arithmetic

  Figure 2.21: Structure of the generic complex-arithmetic system.

(define (make-from-real-imag x y)
  (make-from-real-imag-rectangular x y))
(define (make-from-mag-ang r a)
  (make-from-mag-ang-polar r a))

e resulting complex-number system has the structure shown in Fig-
ure 2.21. e system has been decomposed into three relatively inde-
pendent parts: the complex-number-arithmetic operations, Alyssa’s po-
lar implementation, and Ben’s rectangular implementation. e polar
and rectangular implementations could have been wrien by Ben and
Alyssa working separately, and both of these can be used as underly-
ing representations by a third programmer implementing the complex-
arithmetic procedures in terms of the abstract constructor/selector in-
    Since each data object is tagged with its type, the selectors operate
on the data in a generic manner. at is, each selector is defined to have
a behavior that depends upon the particular type of data it is applied to.

Notice the general mechanism for interfacing the separate representa-
tions: Within a given representation implementation (say, Alyssa’s po-
lar package) a complex number is an untyped pair (magnitude, angle).
When a generic selector operates on a number of polar type, it strips off
the tag and passes the contents on to Alyssa’s code. Conversely, when
Alyssa constructs a number for general use, she tags it with a type so
that it can be appropriately recognized by the higher-level procedures.
is discipline of stripping off and aaching tags as data objects are
passed from level to level can be an important organizational strategy,
as we shall see in Section 2.5.

2.4.3 Data-Directed Programming and Additivity
e general strategy of checking the type of a datum and calling an
appropriate procedure is called dispatching on type. is is a powerful
strategy for obtaining modularity in system design. On the other hand,
implementing the dispatch as in Section 2.4.2 has two significant weak-
nesses. One weakness is that the generic interface procedures (real-
part, imag-part, magnitude, and angle) must know about all the dif-
ferent representations. For instance, suppose we wanted to incorporate
a new representation for complex numbers into our complex-number
system. We would need to identify this new representation with a type,
and then add a clause to each of the generic interface procedures to
check for the new type and apply the appropriate selector for that rep-
    Another weakness of the technique is that even though the indi-
vidual representations can be designed separately, we must guarantee
that no two procedures in the entire system have the same name. is
is why Ben and Alyssa had to change the names of their original proce-
dures from Section 2.4.1.

    e issue underlying both of these weaknesses is that the technique
for implementing generic interfaces is not additive. e person imple-
menting the generic selector procedures must modify those procedures
each time a new representation is installed, and the people interfacing
the individual representations must modify their code to avoid name
conflicts. In each of these cases, the changes that must be made to the
code are straightforward, but they must be made nonetheless, and this
is a source of inconvenience and error. is is not much of a problem
for the complex-number system as it stands, but suppose there were
not two but hundreds of different representations for complex numbers.
And suppose that there were many generic selectors to be maintained
in the abstract-data interface. Suppose, in fact, that no one program-
mer knew all the interface procedures or all the representations. e
problem is real and must be addressed in such programs as large-scale
data-base-management systems.
    What we need is a means for modularizing the system design even
further. is is provided by the programming technique known as data-
directed programming. To understand how data-directed programming
works, begin with the observation that whenever we deal with a set of
generic operations that are common to a set of different types we are,
in effect, dealing with a two-dimensional table that contains the possi-
ble operations on one axis and the possible types on the other axis. e
entries in the table are the procedures that implement each operation
for each type of argument presented. In the complex-number system
developed in the previous section, the correspondence between opera-
tion name, data type, and actual procedure was spread out among the
various conditional clauses in the generic interface procedures. But the
same information could have been organized in a table, as shown in
Figure 2.22.

                                 Polar                  Rectangular

               real-part   real-part-polar         real-part-rectangular

               imag-part   imag-part-polar         imag-part-rectangular
               magnitude   magnitude-polar         magnitude-rectangular
               angle       angle-polar             angle-rectangular

  Figure 2.22: Table of operations for the complex-number system.

    Data-directed programming is the technique of designing programs
to work with such a table directly. Previously, we implemented the
mechanism that interfaces the complex-arithmetic code with the two
representation packages as a set of procedures that each perform an
explicit dispatch on type. Here we will implement the interface as a sin-
gle procedure that looks up the combination of the operation name and
argument type in the table to find the correct procedure to apply, and
then applies it to the contents of the argument. If we do this, then to
add a new representation package to the system we need not change
any existing procedures; we need only add new entries to the table.
    To implement this plan, assume that we have two procedures, put
and get, for manipulating the operation-and-type table:

        • (put ⟨op ⟩ ⟨type ⟩ ⟨item ⟩) installs the ⟨item ⟩ in the table, indexed
          by the ⟨op ⟩ and the ⟨type ⟩.

        • (get ⟨op ⟩ ⟨type ⟩) looks up the ⟨op ⟩, ⟨type ⟩ entry in the table and
          returns the item found there. If no item is found, get returns false.

For now, we can assume that put and get are included in our language.
In Chapter 3 (Section 3.3.3) we will see how to implement these and

other operations for manipulating tables.
    Here is how data-directed programming can be used in the complex-
number system. Ben, who developed the rectangular representation,
implements his code just as he did originally. He defines a collection
of procedures, or a package, and interfaces these to the rest of the sys-
tem by adding entries to the table that tell the system how to operate
on rectangular numbers. is is accomplished by calling the following
(define (install-rectangular-package)
  ;; internal procedures
  (define (real-part z) (car z))
  (define (imag-part z) (cdr z))
  (define (make-from-real-imag x y) (cons x y))
  (define (magnitude z)
    (sqrt (+ (square (real-part z))
                (square (imag-part z)))))
  (define (angle z)
    (atan (imag-part z) (real-part z)))
  (define (make-from-mag-ang r a)
    (cons (* r (cos a)) (* r (sin a))))

  ;; interface to the rest of the system
  (define (tag x) (attach-tag 'rectangular x))
  (put 'real-part '(rectangular) real-part)
  (put 'imag-part '(rectangular) imag-part)
  (put 'magnitude '(rectangular) magnitude)
  (put 'angle '(rectangular) angle)
  (put 'make-from-real-imag 'rectangular
        (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'rectangular
        (lambda (r a) (tag (make-from-mag-ang r a))))

Notice that the internal procedures here are the same procedures from
Section 2.4.1 that Ben wrote when he was working in isolation. No
changes are necessary in order to interface them to the rest of the sys-
tem. Moreover, since these procedure definitions are internal to the in-
stallation procedure, Ben needn’t worry about name conflicts with other
procedures outside the rectangular package. To interface these to the
rest of the system, Ben installs his real-part procedure under the op-
eration name real-part and the type (rectangular), and similarly for
the other selectors.45 e interface also defines the constructors to be
used by the external system.46 ese are identical to Ben’s internally
defined constructors, except that they aach the tag.
    Alyssa’s polar package is analogous:
(define (install-polar-package)
  ;; internal procedures
  (define (magnitude z) (car z))
  (define (angle z) (cdr z))
  (define (make-from-mag-ang r a) (cons r a))
  (define (real-part z) (* (magnitude z) (cos (angle z))))
  (define (imag-part z) (* (magnitude z) (sin (angle z))))
  (define (make-from-real-imag x y)
     (cons (sqrt (+ (square x) (square y)))
             (atan y x)))
  ;; interface to the rest of the system
  (define (tag x) (attach-tag 'polar x))
  (put 'real-part '(polar) real-part)
  (put 'imag-part '(polar) imag-part)
  (put 'magnitude '(polar) magnitude)

  45 We  use the list (rectangular) rather than the symbol rectangular to allow for
the possibility of operations with multiple arguments, not all of the same type.
   46 e type the constructors are installed under needn’t be a list because a constructor

is always used to make an object of one particular type.

  (put 'angle '(polar) angle)
  (put 'make-from-real-imag 'polar
         (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'polar
         (lambda (r a) (tag (make-from-mag-ang r a))))

Even though Ben and Alyssa both still use their original procedures
defined with the same names as each other’s (e.g., real-part), these
definitions are now internal to different procedures (see Section 1.1.8),
so there is no name conflict.
    e complex-arithmetic selectors access the table by means of a
general “operation” procedure called apply-generic, which applies a
generic operation to some arguments. apply-generic looks in the ta-
ble under the name of the operation and the types of the arguments and
applies the resulting procedure if one is present:47
(define (apply-generic op . args)
  (let ((type-tags (map type-tag args)))
     (let ((proc (get op type-tags)))
        (if proc
              (apply proc (map contents args))

  47 apply-generic uses the doed-tail notation described in Exercise 2.20, because dif-

ferent generic operations may take different numbers of arguments. In apply-generic,
op has as its value the first argument to apply-generic and args has as its value a list
of the remaining arguments.
   apply-generic also uses the primitive procedure apply, which takes two arguments,
a procedure and a list. apply applies the procedure, using the elements in the list as
arguments. For example,
(apply + (list 1 2 3 4))

returns 10.

             "No method for these types: APPLY-GENERIC"
             (list op type-tags))))))

Using apply-generic, we can define our generic selectors as follows:
(define (real-part z) (apply-generic 'real-part z))
(define (imag-part z) (apply-generic 'imag-part z))
(define (magnitude z) (apply-generic 'magnitude z))
(define (angle z) (apply-generic 'angle z))

Observe that these do not change at all if a new representation is added
to the system.
    We can also extract from the table the constructors to be used by the
programs external to the packages in making complex numbers from
real and imaginary parts and from magnitudes and angles. As in Section
2.4.2, we construct rectangular numbers whenever we have real and
imaginary parts, and polar numbers whenever we have magnitudes and
(define (make-from-real-imag x y)
  ((get 'make-from-real-imag 'rectangular) x y))
(define (make-from-mag-ang r a)
  ((get 'make-from-mag-ang 'polar) r a))

      Exercise 2.73: Section 2.3.2 described a program that per-
      forms symbolic differentiation:
      (define (deriv exp var)
        (cond ((number? exp) 0)
               ((variable? exp)
                (if (same-variable? exp var) 1 0))
               ((sum? exp)
                (make-sum (deriv (addend exp) var)
                           (deriv (augend exp) var)))

         ((product? exp)
          (make-sum (make-product
                      (multiplier exp)
                      (deriv (multiplicand exp) var))
                      (deriv (multiplier exp) var)
                      (multiplicand exp))))
         ⟨more rules can be added here⟩
         (else (error "unknown expression type:
                         DERIV" exp))))

We can regard this program as performing a dispatch on
the type of the expression to be differentiated. In this situ-
ation the “type tag” of the datum is the algebraic operator
symbol (such as +) and the operation being performed is
deriv. We can transform this program into data-directed
style by rewriting the basic derivative procedure as
(define (deriv exp var)
  (cond ((number? exp) 0)
         ((variable? exp) (if (same-variable? exp var) 1 0))
         (else ((get 'deriv (operator exp))
                 (operands exp) var))))
(define (operator exp) (car exp))
(define (operands exp) (cdr exp))

  a. Explain what was done above. Why can’t we assim-
     ilate the predicates number? and variable? into the
     data-directed dispatch?
  b. Write the procedures for derivatives of sums and prod-
     ucts, and the auxiliary code required to install them in
     the table used by the program above.

  c. Choose any additional differentiation rule that you
     like, such as the one for exponents (Exercise 2.56), and
     install it in this data-directed system.
  d. In this simple algebraic manipulator the type of an
     expression is the algebraic operator that binds it to-
     gether. Suppose, however, we indexed the procedures
     in the opposite way, so that the dispatch line in deriv
     looked like
     ((get (operator exp) 'deriv) (operands exp) var)

     What corresponding changes to the derivative system
     are required?

Exercise 2.74: Insatiable Enterprises, Inc., is a highly de-
centralized conglomerate company consisting of a large num-
ber of independent divisions located all over the world. e
company’s computer facilities have just been interconnected
by means of a clever network-interfacing scheme that makes
the entire network appear to any user to be a single com-
puter. Insatiable’s president, in her first aempt to exploit
the ability of the network to extract administrative infor-
mation from division files, is dismayed to discover that, al-
though all the division files have been implemented as data
structures in Scheme, the particular data structure used varies
from division to division. A meeting of division managers
is hastily called to search for a strategy to integrate the files
that will satisfy headquarters’ needs while preserving the
existing autonomy of the divisions.
Show how such a strategy can be implemented with data-
directed programming. As an example, suppose that each

division’s personnel records consist of a single file, which
contains a set of records keyed on employees’ names. e
structure of the set varies from division to division. Fur-
thermore, each employee’s record is itself a set (structured
differently from division to division) that contains informa-
tion keyed under identifiers such as address and salary.
In particular:

  a. Implement for headquarters a get-record procedure
     that retrieves a specified employee’s record from a
     specified personnel file. e procedure should be ap-
     plicable to any division’s file. Explain how the individ-
     ual divisions’ files should be structured. In particular,
     what type information must be supplied?
  b. Implement for headquarters a get-salary procedure
     that returns the salary information from a given em-
     ployee’s record from any division’s personnel file. How
     should the record be structured in order to make this
     operation work?
  c. Implement for headquarters a find-employee-record
     procedure. is should search all the divisions’ files
     for the record of a given employee and return the record.
     Assume that this procedure takes as arguments an
     employee’s name and a list of all the divisions’ files.
  d. When Insatiable takes over a new company, what changes
     must be made in order to incorporate the new person-
     nel information into the central system?

Message passing
e key idea of data-directed programming is to handle generic opera-
tions in programs by dealing explicitly with operation-and-type tables,
such as the table in Figure 2.22. e style of programming we used in
Section 2.4.2 organized the required dispatching on type by having each
operation take care of its own dispatching. In effect, this decomposes the
operation-and-type table into rows, with each generic operation proce-
dure representing a row of the table.
    An alternative implementation strategy is to decompose the table
into columns and, instead of using “intelligent operations” that dispatch
on data types, to work with “intelligent data objects” that dispatch on
operation names. We can do this by arranging things so that a data
object, such as a rectangular number, is represented as a procedure that
takes as input the required operation name and performs the operation
indicated. In such a discipline, make-from-real-imag could be wrien
(define (make-from-real-imag x y)
  (define (dispatch op)
     (cond ((eq? op 'real-part) x)
            ((eq? op 'imag-part) y)
            ((eq? op 'magnitude) (sqrt (+ (square x) (square y))))
            ((eq? op 'angle) (atan y x))
            (else (error "Unknown op: MAKE-FROM-REAL-IMAG" op))))

e corresponding apply-generic procedure, which applies a generic
operation to an argument, now simply feeds the operation’s name to
the data object and lets the object do the work:48
  48 Onelimitation of this organization is it permits only generic procedures of one

(define (apply-generic op arg) (arg op))

Note that the value returned by make-from-real-imag is a procedure—
the internal dispatch procedure. is is the procedure that is invoked
when apply-generic requests an operation to be performed.
    is style of programming is called message passing. e name comes
from the image that a data object is an entity that receives the requested
operation name as a “message.” We have already seen an example of
message passing in Section 2.1.3, where we saw how cons, car, and cdr
could be defined with no data objects but only procedures. Here we see
that message passing is not a mathematical trick but a useful technique
for organizing systems with generic operations. In the remainder of this
chapter we will continue to use data-directed programming, rather than
message passing, to discuss generic arithmetic operations. In Chapter 3
we will return to message passing, and we will see that it can be a pow-
erful tool for structuring simulation programs.

      Exercise 2.75: Implement the constructor make-from-mag-
      ang in message-passing style. is procedure should be anal-
      ogous to the make-from-real-imag procedure given above.

      Exercise 2.76: As a large system with generic operations
      evolves, new types of data objects or new operations may
      be needed. For each of the three strategies—generic opera-
      tions with explicit dispatch, data-directed style, and message-
      passing-style—describe the changes that must be made to a
      system in order to add new types or new operations. Which
      organization would be most appropriate for a system in
      which new types must oen be added? Which would be
      most appropriate for a system in which new operations
      must oen be added?

2.5 Systems with Generic Operations
In the previous section, we saw how to design systems in which data
objects can be represented in more than one way. e key idea is to
link the code that specifies the data operations to the several represen-
tations by means of generic interface procedures. Now we will see how
to use this same idea not only to define operations that are generic over
different representations but also to define operations that are generic
over different kinds of arguments. We have already seen several dif-
ferent packages of arithmetic operations: the primitive arithmetic (+, -,
*, /) built into our language, the rational-number arithmetic (add-rat,
sub-rat, mul-rat, div-rat) of Section 2.1.1, and the complex-number
arithmetic that we implemented in Section 2.4.3. We will now use data-
directed techniques to construct a package of arithmetic operations that
incorporates all the arithmetic packages we have already constructed.
     Figure 2.23 shows the structure of the system we shall build. Notice
the abstraction barriers. From the perspective of someone using “num-
bers,” there is a single procedure add that operates on whatever num-
bers are supplied. add is part of a generic interface that allows the sep-
arate ordinary-arithmetic, rational-arithmetic, and complex-arithmetic
packages to be accessed uniformly by programs that use numbers. Any
individual arithmetic package (such as the complex package) may it-
self be accessed through generic procedures (such as add-complex) that
combine packages designed for different representations (such as rect-
angular and polar). Moreover, the structure of the system is additive, so
that one can design the individual arithmetic packages separately and
combine them to produce a generic arithmetic system.

                               Programs that use numbers

                                  add    sub   mul   div

                               Generic arithmetic package

    add-rat sub-rat            add-complex sub-complex
                                                                        + -- * /
    mul-rat div-rat            mul-complex div-complex

                                   Complex arithmetic
        Rational                                                         Ordinary
       arithmetic                                                       arithmetic
                             Rectangular             Polar

                      List structure and primitive machine arithmetic

                Figure 2.23: Generic arithmetic system.

2.5.1 Generic Arithmetic Operations
e task of designing generic arithmetic operations is analogous to that
of designing the generic complex-number operations. We would like,
for instance, to have a generic addition procedure add that acts like or-
dinary primitive addition + on ordinary numbers, like add-rat on ra-
tional numbers, and like add-complex on complex numbers. We can
implement add, and the other generic arithmetic operations, by follow-
ing the same strategy we used in Section 2.4.3 to implement the generic
selectors for complex numbers. We will aach a type tag to each kind of
number and cause the generic procedure to dispatch to an appropriate
package according to the data type of its arguments.
    e generic arithmetic procedures are defined as follows:
(define (add x y) (apply-generic 'add x y))

(define (sub x y) (apply-generic 'sub x y))
(define (mul x y) (apply-generic 'mul x y))
(define (div x y) (apply-generic 'div x y))

We begin by installing a package for handling ordinary numbers, that
is, the primitive numbers of our language. We will tag these with the
symbol scheme-number. e arithmetic operations in this package are
the primitive arithmetic procedures (so there is no need to define extra
procedures to handle the untagged numbers). Since these operations
each take two arguments, they are installed in the table keyed by the
list (scheme-number scheme-number):
(define (install-scheme-number-package)
  (define (tag x) (attach-tag 'scheme-number x))
  (put 'add '(scheme-number scheme-number)
       (lambda (x y) (tag (+ x y))))
  (put 'sub '(scheme-number scheme-number)
       (lambda (x y) (tag (- x y))))
  (put 'mul '(scheme-number scheme-number)
       (lambda (x y) (tag (* x y))))
  (put 'div '(scheme-number scheme-number)
       (lambda (x y) (tag (/ x y))))
  (put 'make 'scheme-number (lambda (x) (tag x)))

Users of the Scheme-number package will create (tagged) ordinary num-
bers by means of the procedure:
(define (make-scheme-number n)
  ((get 'make 'scheme-number) n))

Now that the framework of the generic arithmetic system is in place,
we can readily include new kinds of numbers. Here is a package that
performs rational arithmetic. Notice that, as a benefit of additivity, we

can use without modification the rational-number code from Section
2.1.1 as the internal procedures in the package:
(define (install-rational-package)
  ;; internal procedures
  (define (numer x) (car x))
  (define (denom x) (cdr x))
  (define (make-rat n d)
    (let ((g (gcd n d)))
       (cons (/ n g) (/ d g))))
  (define (add-rat x y)
    (make-rat (+ (* (numer x) (denom y))
                     (* (numer y) (denom x)))
                 (* (denom x) (denom y))))
  (define (sub-rat x y)
    (make-rat (- (* (numer x) (denom y))
                     (* (numer y) (denom x)))
                 (* (denom x) (denom y))))
  (define (mul-rat x y)
    (make-rat (* (numer x) (numer y))
                 (* (denom x) (denom y))))
  (define (div-rat x y)
    (make-rat (* (numer x) (denom y))
                 (* (denom x) (numer y))))
  ;; interface to rest of the system
  (define (tag x) (attach-tag 'rational x))
  (put 'add '(rational rational)
        (lambda (x y) (tag (add-rat x y))))
  (put 'sub '(rational rational)
        (lambda (x y) (tag (sub-rat x y))))
  (put 'mul '(rational rational)
        (lambda (x y) (tag (mul-rat x y))))
  (put 'div '(rational rational)
        (lambda (x y) (tag (div-rat x y))))

  (put 'make 'rational
        (lambda (n d) (tag (make-rat n d))))
(define (make-rational n d)
  ((get 'make 'rational) n d))

We can install a similar package to handle complex numbers, using the
tag complex. In creating the package, we extract from the table the op-
erations make-from-real-imag and make-from-mag-ang that were de-
fined by the rectangular and polar packages. Additivity permits us to
use, as the internal operations, the same add-complex, sub-complex,
mul-complex, and div-complex procedures from Section 2.4.1.
(define (install-complex-package)
  ;; imported procedures from rectangular and polar packages
  (define (make-from-real-imag x y)
    ((get 'make-from-real-imag 'rectangular) x y))
  (define (make-from-mag-ang r a)
    ((get 'make-from-mag-ang 'polar) r a))
  ;; internal procedures
  (define (add-complex z1 z2)
    (make-from-real-imag (+ (real-part z1) (real-part z2))
                               (+ (imag-part z1) (imag-part z2))))
  (define (sub-complex z1 z2)
    (make-from-real-imag (- (real-part z1) (real-part z2))
                               (- (imag-part z1) (imag-part z2))))
  (define (mul-complex z1 z2)
    (make-from-mag-ang (* (magnitude z1) (magnitude z2))
                             (+ (angle z1) (angle z2))))
  (define (div-complex z1 z2)
    (make-from-mag-ang (/ (magnitude z1) (magnitude z2))
                             (- (angle z1) (angle z2))))
  ;; interface to rest of the system
  (define (tag z) (attach-tag 'complex z))

  (put 'add '(complex complex)
       (lambda (z1 z2) (tag (add-complex z1 z2))))
  (put 'sub '(complex complex)
       (lambda (z1 z2) (tag (sub-complex z1 z2))))
  (put 'mul '(complex complex)
       (lambda (z1 z2) (tag (mul-complex z1 z2))))
  (put 'div '(complex complex)
       (lambda (z1 z2) (tag (div-complex z1 z2))))
  (put 'make-from-real-imag 'complex
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'complex
       (lambda (r a) (tag (make-from-mag-ang r a))))

Programs outside the complex-number package can construct complex
numbers either from real and imaginary parts or from magnitudes and
angles. Notice how the underlying procedures, originally defined in the
rectangular and polar packages, are exported to the complex package,
and exported from there to the outside world.
(define (make-complex-from-real-imag x y)
  ((get 'make-from-real-imag 'complex) x y))
(define (make-complex-from-mag-ang r a)
  ((get 'make-from-mag-ang 'complex) r a))

What we have here is a two-level tag system. A typical complex num-
ber, such as 3 + 4i in rectangular form, would be represented as shown
in Figure 2.24. e outer tag (complex) is used to direct the number to
the complex package. Once within the complex package, the next tag
(rectangular) is used to direct the number to the rectangular package.
In a large and complicated system there might be many levels, each in-
terfaced with the next by means of generic operations. As a data object
is passed “downward,” the outer tag that is used to direct it to the ap-

                     complex     rectangular     3    4

      Figure 2.24: Representation of 3 + 4i in rectangular form.

propriate package is stripped off (by applying contents) and the next
level of tag (if any) becomes visible to be used for further dispatching.
    In the above packages, we used add-rat, add-complex, and the other
arithmetic procedures exactly as originally wrien. Once these defini-
tions are internal to different installation procedures, however, they no
longer need names that are distinct from each other: we could simply
name them add, sub, mul, and div in both packages.

      Exercise 2.77: Louis Reasoner tries to evaluate the expres-
      sion (magnitude z) where z is the object shown in Figure
      2.24. To his surprise, instead of the answer 5 he gets an error
      message from apply-generic, saying there is no method
      for the operation magnitude on the types (complex). He
      shows this interaction to Alyssa P. Hacker, who says “e
      problem is that the complex-number selectors were never
      defined for complex numbers, just for polar and rectangular
      numbers. All you have to do to make this work is add the
      following to the complex package:”
      (put 'real-part '(complex) real-part)
      (put 'imag-part '(complex) imag-part)
      (put 'magnitude '(complex) magnitude)
      (put 'angle '(complex) angle)

Describe in detail why this works. As an example, trace
through all the procedures called in evaluating the expres-
sion (magnitude z) where z is the object shown in Figure
2.24. In particular, how many times is apply-generic in-
voked? What procedure is dispatched to in each case?

Exercise 2.78: e internal procedures in the scheme-number
package are essentially nothing more than calls to the prim-
itive procedures +, -, etc. It was not possible to use the prim-
itives of the language directly because our type-tag system
requires that each data object have a type aached to it. In
fact, however, all Lisp implementations do have a type sys-
tem, which they use internally. Primitive predicates such
as symbol? and number? determine whether data objects
have particular types. Modify the definitions of type-tag,
contents, and attach-tag from Section 2.4.2 so that our
generic system takes advantage of Scheme’s internal type
system. at is to say, the system should work as before ex-
cept that ordinary numbers should be represented simply
as Scheme numbers rather than as pairs whose car is the
symbol scheme-number.

Exercise 2.79: Define a generic equality predicate equ? that
tests the equality of two numbers, and install it in the generic
arithmetic package. is operation should work for ordi-
nary numbers, rational numbers, and complex numbers.

Exercise 2.80: Define a generic predicate =zero? that tests
if its argument is zero, and install it in the generic arith-
metic package. is operation should work for ordinary
numbers, rational numbers, and complex numbers.

2.5.2 Combining Data of Different Types
We have seen how to define a unified arithmetic system that encom-
passes ordinary numbers, complex numbers, rational numbers, and any
other type of number we might decide to invent, but we have ignored an
important issue. e operations we have defined so far treat the differ-
ent data types as being completely independent. us, there are separate
packages for adding, say, two ordinary numbers, or two complex num-
bers. What we have not yet considered is the fact that it is meaningful to
define operations that cross the type boundaries, such as the addition of
a complex number to an ordinary number. We have gone to great pains
to introduce barriers between parts of our programs so that they can be
developed and understood separately. We would like to introduce the
cross-type operations in some carefully controlled way, so that we can
support them without seriously violating our module boundaries.
    One way to handle cross-type operations is to design a different pro-
cedure for each possible combination of types for which the operation
is valid. For example, we could extend the complex-number package so
that it provides a procedure for adding complex numbers to ordinary
numbers and installs this in the table using the tag (complex scheme-
;; to be included in the complex package
(define (add-complex-to-schemenum z x)
  (make-from-real-imag (+ (real-part z) x) (imag-part z)))
(put 'add '(complex scheme-number)
      (lambda (z x) (tag (add-complex-to-schemenum z x))))

is technique works, but it is cumbersome. With such a system, the
cost of introducing a new type is not just the construction of the pack-
  49 We also have to supply an almost identical procedure to handle the types (scheme-

number complex).

age of procedures for that type but also the construction and installa-
tion of the procedures that implement the cross-type operations. is
can easily be much more code than is needed to define the operations
on the type itself. e method also undermines our ability to combine
separate packages additively, or at least to limit the extent to which the
implementors of the individual packages need to take account of other
packages. For instance, in the example above, it seems reasonable that
handling mixed operations on complex numbers and ordinary numbers
should be the responsibility of the complex-number package. Combin-
ing rational numbers and complex numbers, however, might be done by
the complex package, by the rational package, or by some third package
that uses operations extracted from these two packages. Formulating
coherent policies on the division of responsibility among packages can
be an overwhelming task in designing systems with many packages and
many cross-type operations.

In the general situation of completely unrelated operations acting on
completely unrelated types, implementing explicit cross-type operations,
cumbersome though it may be, is the best that one can hope for. For-
tunately, we can usually do beer by taking advantage of additional
structure that may be latent in our type system. Oen the different data
types are not completely independent, and there may be ways by which
objects of one type may be viewed as being of another type. is process
is called coercion. For example, if we are asked to arithmetically combine
an ordinary number with a complex number, we can view the ordinary
number as a complex number whose imaginary part is zero. is trans-
forms the problem to that of combining two complex numbers, which
can be handled in the ordinary way by the complex-arithmetic package.

    In general, we can implement this idea by designing coercion pro-
cedures that transform an object of one type into an equivalent object
of another type. Here is a typical coercion procedure, which transforms
a given ordinary number to a complex number with that real part and
zero imaginary part:
(define (scheme-number->complex n)
  (make-complex-from-real-imag (contents n) 0))

We install these coercion procedures in a special coercion table, indexed
under the names of the two types:
(put-coercion 'scheme-number

(We assume that there are put-coercion and get-coercion procedures
available for manipulating this table.) Generally some of the slots in the
table will be empty, because it is not generally possible to coerce an ar-
bitrary data object of each type into all other types. For example, there
is no way to coerce an arbitrary complex number to an ordinary num-
ber, so there will be no general complex->scheme-number procedure
included in the table.
    Once the coercion table has been set up, we can handle coercion
in a uniform manner by modifying the apply-generic procedure of
Section 2.4.3. When asked to apply an operation, we first check whether
the operation is defined for the arguments’ types, just as before. If so,
we dispatch to the procedure found in the operation-and-type table.
Otherwise, we try coercion. For simplicity, we consider only the case
where there are two arguments.50 We check the coercion table to see
if objects of the first type can be coerced to the second type. If so, we
  50 See   Exercise 2.82 for generalizations.

coerce the first argument and try the operation again. If objects of the
first type cannot in general be coerced to the second type, we try the
coercion the other way around to see if there is a way to coerce the
second argument to the type of the first argument. Finally, if there is no
known way to coerce either type to the other type, we give up. Here is
the procedure:
(define (apply-generic op . args)
  (let ((type-tags (map type-tag args)))
    (let ((proc (get op type-tags)))
      (if proc
           (apply proc (map contents args))
           (if (= (length args) 2)
                 (let ((type1 (car type-tags))
                      (type2 (cadr type-tags))
                      (a1 (car args))
                      (a2 (cadr args)))
                  (let ((t1->t2 (get-coercion type1 type2))
                        (t2->t1 (get-coercion type2 type1)))
                    (cond (t1->t2
                            (apply-generic op (t1->t2 a1) a2))
                            (apply-generic op a1 (t2->t1 a2)))
                           (else (error "No method for these types"
                                         (list op type-tags))))))
                 (error "No method for these types"
                       (list op type-tags)))))))

is coercion scheme has many advantages over the method of defining
explicit cross-type operations, as outlined above. Although we still need
to write coercion procedures to relate the types (possibly n 2 procedures
for a system with n types), we need to write only one procedure for
each pair of types rather than a different procedure for each collection

of types and each generic operation.51 What we are counting on here
is the fact that the appropriate transformation between types depends
only on the types themselves, not on the operation to be applied.
     On the other hand, there may be applications for which our coer-
cion scheme is not general enough. Even when neither of the objects to
be combined can be converted to the type of the other it may still be
possible to perform the operation by converting both objects to a third
type. In order to deal with such complexity and still preserve modular-
ity in our programs, it is usually necessary to build systems that take
advantage of still further structure in the relations among types, as we
discuss next.

Hierarchies of types
e coercion scheme presented above relied on the existence of natural
relations between pairs of types. Oen there is more “global” structure
in how the different types relate to each other. For instance, suppose
we are building a generic arithmetic system to handle integers, rational
numbers, real numbers, and complex numbers. In such a system, it is
quite natural to regard an integer as a special kind of rational number,
which is in turn a special kind of real number, which is in turn a special
kind of complex number. What we actually have is a so-called hierarchy
of types, in which, for example, integers are a subtype of rational num-
  51 If we are clever, we can usually get by with fewer than n 2 coercion procedures.

For instance, if we know how to convert from type 1 to type 2 and from type 2 to
type 3, then we can use this knowledge to convert from type 1 to type 3. is can
greatly decrease the number of coercion procedures we need to supply explicitly when
we add a new type to the system. If we are willing to build the required amount of
sophistication into our system, we can have it search the “graph” of relations among
types and automatically generate those coercion procedures that can be inferred from
the ones that are supplied explicitly.





                    Figure 2.25: A tower of types.

bers (i.e., any operation that can be applied to a rational number can
automatically be applied to an integer). Conversely, we say that ratio-
nal numbers form a supertype of integers. e particular hierarchy we
have here is of a very simple kind, in which each type has at most one
supertype and at most one subtype. Such a structure, called a tower, is
illustrated in Figure 2.25.
     If we have a tower structure, then we can greatly simplify the prob-
lem of adding a new type to the hierarchy, for we need only specify
how the new type is embedded in the next supertype above it and how
it is the supertype of the type below it. For example, if we want to add
an integer to a complex number, we need not explicitly define a special
coercion procedure integer->complex. Instead, we define how an inte-
ger can be transformed into a rational number, how a rational number is
transformed into a real number, and how a real number is transformed
into a complex number. We then allow the system to transform the in-
teger into a complex number through these steps and then add the two
complex numbers.
     We can redesign our apply-generic procedure in the following
way: For each type, we need to supply a raise procedure, which “raises”

objects of that type one level in the tower. en when the system is re-
quired to operate on objects of different types it can successively raise
the lower types until all the objects are at the same level in the tower.
(Exercise 2.83 and Exercise 2.84 concern the details of implementing
such a strategy.)
     Another advantage of a tower is that we can easily implement the
notion that every type “inherits” all operations defined on a supertype.
For instance, if we do not supply a special procedure for finding the real
part of an integer, we should nevertheless expect that real-part will
be defined for integers by virtue of the fact that integers are a subtype
of complex numbers. In a tower, we can arrange for this to happen in a
uniform way by modifying apply-generic. If the required operation is
not directly defined for the type of the object given, we raise the object
to its supertype and try again. We thus crawl up the tower, transforming
our argument as we go, until we either find a level at which the desired
operation can be performed or hit the top (in which case we give up).
     Yet another advantage of a tower over a more general hierarchy is
that it gives us a simple way to “lower” a data object to the simplest
representation. For example, if we add 2 + 3i to 4 − 3i, it would be nice
to obtain the answer as the integer 6 rather than as the complex num-
ber 6 + 0i. Exercise 2.85 discusses a way to implement such a lowering
operation. (e trick is that we need a general way to distinguish those
objects that can be lowered, such as 6 + 0i, from those that cannot, such
as 6 + 2i.)

Inadequacies of hierarchies
If the data types in our system can be naturally arranged in a tower,
this greatly simplifies the problems of dealing with generic operations
on different types, as we have seen. Unfortunately, this is usually not the
case. Figure 2.26 illustrates a more complex arrangement of mixed types,


                               triangle              trapezoid

                   isosceles                right         parallelogram           kite
                    triangle              triangle
                                                     rectangle          rhombus

     equilateral             isosceles
      triangle            right triangle                       square

       Figure 2.26: Relations among types of geometric figures.

this one showing relations among different types of geometric figures.
We see that, in general, a type may have more than one subtype. Tri-
angles and quadrilaterals, for instance, are both subtypes of polygons.
In addition, a type may have more than one supertype. For example,
an isosceles right triangle may be regarded either as an isosceles trian-
gle or as a right triangle. is multiple-supertypes issue is particularly
thorny, since it means that there is no unique way to “raise” a type in the
hierarchy. Finding the “correct” supertype in which to apply an opera-
tion to an object may involve considerable searching through the entire
type network on the part of a procedure such as apply-generic. Since
there generally are multiple subtypes for a type, there is a similar prob-
lem in coercing a value “down” the type hierarchy. Dealing with large
numbers of interrelated types while still preserving modularity in the

design of large systems is very difficult, and is an area of much current

       Exercise 2.81: Louis Reasoner has noticed that apply-generic
       may try to coerce the arguments to each other’s type even
       if they already have the same type. erefore, he reasons,
       we need to put procedures in the coercion table to coerce
       arguments of each type to their own type. For example, in
       addition to the scheme-number->complex coercion shown
       above, he would do:
       (define (scheme-number->scheme-number n) n)
       (define (complex->complex z) z)
       (put-coercion 'scheme-number
       (put-coercion 'complex 'complex complex->complex)

  52 is  statement, which also appears in the first edition of this book, is just as true
now as it was when we wrote it twelve years ago. Developing a useful, general frame-
work for expressing the relations among different types of entities (what philosophers
call “ontology”) seems intractably difficult. e main difference between the confu-
sion that existed ten years ago and the confusion that exists now is that now a va-
riety of inadequate ontological theories have been embodied in a plethora of corre-
spondingly inadequate programming languages. For example, much of the complexity
of object-oriented programming languages—and the subtle and confusing differences
among contemporary object-oriented languages—centers on the treatment of generic
operations on interrelated types. Our own discussion of computational objects in Chap-
ter 3 avoids these issues entirely. Readers familiar with object-oriented programming
will notice that we have much to say in chapter 3 about local state, but we do not even
mention “classes” or “inheritance.” In fact, we suspect that these problems cannot be ad-
equately addressed in terms of computer-language design alone, without also drawing
on work in knowledge representation and automated reasoning.

  a. With Louis’s coercion procedures installed, what hap-
     pens if apply-generic is called with two arguments
     of type scheme-number or two arguments of type complex
     for an operation that is not found in the table for those
     types? For example, assume that we’ve defined a generic
     exponentiation operation:
     (define (exp x y) (apply-generic 'exp x y))

     and have put a procedure for exponentiation in the
     Scheme-number package but not in any other pack-
     ;; following added to Scheme-number package
     (put 'exp '(scheme-number scheme-number)
          (lambda (x y) (tag (expt x y))))
          ; using primitive expt

     What happens if we call exp with two complex num-
     bers as arguments?
  b. Is Louis correct that something had to be done about
     coercion with arguments of the same type, or does
     apply-generic work correctly as is?

  c. Modify apply-generic so that it doesn’t try coercion
     if the two arguments have the same type.

Exercise 2.82: Show how to generalize apply-generic to
handle coercion in the general case of multiple arguments.
One strategy is to aempt to coerce all the arguments to
the type of the first argument, then to the type of the sec-
ond argument, and so on. Give an example of a situation

where this strategy (and likewise the two-argument ver-
sion given above) is not sufficiently general. (Hint: Con-
sider the case where there are some suitable mixed-type
operations present in the table that will not be tried.)

Exercise 2.83: Suppose you are designing a generic arith-
metic system for dealing with the tower of types shown in
Figure 2.25: integer, rational, real, complex. For each type
(except complex), design a procedure that raises objects of
that type one level in the tower. Show how to install a
generic raise operation that will work for each type (ex-
cept complex).

Exercise 2.84: Using the raise operation of Exercise 2.83,
modify the apply-generic procedure so that it coerces its
arguments to have the same type by the method of succes-
sive raising, as discussed in this section. You will need to
devise a way to test which of two types is higher in the
tower. Do this in a manner that is “compatible” with the
rest of the system and will not lead to problems in adding
new levels to the tower.

Exercise 2.85: is section mentioned a method for “sim-
plifying” a data object by lowering it in the tower of types
as far as possible. Design a procedure drop that accom-
plishes this for the tower described in Exercise 2.83. e
key is to decide, in some general way, whether an object
can be lowered. For example, the complex number 1.5 + 0i
can be lowered as far as real, the complex number 1 + 0i
can be lowered as far as integer, and the complex number

         2 + 3i cannot be lowered at all. Here is a plan for determin-
         ing whether an object can be lowered: Begin by defining
         a generic operation project that “pushes” an object down
         in the tower. For example, projecting a complex number
         would involve throwing away the imaginary part. en a
         number can be dropped if, when we project it and raise
         the result back to the type we started with, we end up with
         something equal to what we started with. Show how to im-
         plement this idea in detail, by writing a drop procedure that
         drops an object as far as possible. You will need to design
         the various projection operations53 and install project as a
         generic operation in the system. You will also need to make
         use of a generic equality predicate, such as described in
         Exercise 2.79. Finally, use drop to rewrite apply-generic
         from Exercise 2.84 so that it “simplifies” its answers.

         Exercise 2.86: Suppose we want to handle complex num-
         bers whose real parts, imaginary parts, magnitudes, and an-
         gles can be either ordinary numbers, rational numbers, or
         other numbers we might wish to add to the system. De-
         scribe and implement the changes to the system needed to
         accommodate this. You will have to define operations such
         as sine and cosine that are generic over ordinary numbers
         and rational numbers.

  53 A real number can be projected to an integer using the round primitive, which
returns the closest integer to its argument.

2.5.3 Example: Symbolic Algebra
e manipulation of symbolic algebraic expressions is a complex pro-
cess that illustrates many of the hardest problems that occur in the de-
sign of large-scale systems. An algebraic expression, in general, can
be viewed as a hierarchical structure, a tree of operators applied to
operands. We can construct algebraic expressions by starting with a
set of primitive objects, such as constants and variables, and combining
these by means of algebraic operators, such as addition and multipli-
cation. As in other languages, we form abstractions that enable us to
refer to compound objects in simple terms. Typical abstractions in sym-
bolic algebra are ideas such as linear combination, polynomial, rational
function, or trigonometric function. We can regard these as compound
“types,” which are oen useful for directing the processing of expres-
sions. For example, we could describe the expression
                x 2 sin(y 2 + 1) + x cos 2y + cos(y 3 − 2y 2 )
as a polynomial in x with coefficients that are trigonometric functions
of polynomials in y whose coefficients are integers.
     We will not aempt to develop a complete algebraic-manipulation
system here. Such systems are exceedingly complex programs, embody-
ing deep algebraic knowledge and elegant algorithms. What we will do
is look at a simple but important part of algebraic manipulation: the
arithmetic of polynomials. We will illustrate the kinds of decisions the
designer of such a system faces, and how to apply the ideas of abstract
data and generic operations to help organize this effort.

Arithmetic on polynomials
Our first task in designing a system for performing arithmetic on poly-
nomials is to decide just what a polynomial is. Polynomials are normally

defined relative to certain variables (the indeterminates of the polyno-
mial). For simplicity, we will restrict ourselves to polynomials having
just one indeterminate (univariate polynomials).54 We will define a poly-
nomial to be a sum of terms, each of which is either a coefficient, a
power of the indeterminate, or a product of a coefficient and a power
of the indeterminate. A coefficient is defined as an algebraic expression
that is not dependent upon the indeterminate of the polynomial. For
                              5x 2 + 3x + 7
is a simple polynomial in x, and

                                (y 2 + 1)x 3 + (2y)x + 1

is a polynomial in x whose coefficients are polynomials in y.
     Already we are skirting some thorny issues. Is the first of these poly-
nomials the same as the polynomial 5y 2 + 3y + 7, or not? A reasonable
answer might be “yes, if we are considering a polynomial purely as a
mathematical function, but no, if we are considering a polynomial to
be a syntactic form.” e second polynomial is algebraically equivalent
to a polynomial in y whose coefficients are polynomials in x . Should
our system recognize this, or not? Furthermore, there are other ways to
represent a polynomial—for example, as a product of factors, or (for a
univariate polynomial) as the set of roots, or as a listing of the values of
the polynomial at a specified set of points.55 We can finesse these ques-
  54 On   the other hand, we will allow polynomials whose coefficients are themselves
polynomials in other variables. is will give us essentially the same representational
power as a full multivariate system, although it does lead to coercion problems, as
discussed below.
   55 For univariate polynomials, giving the value of a polynomial at a given set of points

can be a particularly good representation. is makes polynomial arithmetic extremely

tions by deciding that in our algebraic-manipulation system a “polyno-
mial” will be a particular syntactic form, not its underlying mathemat-
ical meaning.
     Now we must consider how to go about doing arithmetic on polyno-
mials. In this simple system, we will consider only addition and multi-
plication. Moreover, we will insist that two polynomials to be combined
must have the same indeterminate.
     We will approach the design of our system by following the familiar
discipline of data abstraction. We will represent polynomials using a
data structure called a poly, which consists of a variable and a collection
of terms. We assume that we have selectors variable and term-list
that extract those parts from a poly and a constructor make-poly that
assembles a poly from a given variable and a term list. A variable will be
just a symbol, so we can use the same-variable? procedure of Section
2.3.2 to compare variables. e following procedures define addition and
multiplication of polys:
(define (add-poly p1 p2)
  (if (same-variable? (variable p1) (variable p2))
       (make-poly (variable p1)
                      (add-terms (term-list p1) (term-list p2)))
       (error "Polys not in same var: ADD-POLY" (list p1 p2))))
(define (mul-poly p1 p2)
  (if (same-variable? (variable p1) (variable p2))
       (make-poly (variable p1)
                      (mul-terms (term-list p1) (term-list p2)))
       (error "Polys not in same var: MUL-POLY" (list p1 p2))))

simple. To obtain, for example, the sum of two polynomials represented in this way,
we need only add the values of the polynomials at corresponding points. To transform
back to a more familiar representation, we can use the Lagrange interpolation formula,
which shows how to recover the coefficients of a polynomial of degree n given the
values of the polynomial at n + 1 points.

To incorporate polynomials into our generic arithmetic system, we need
to supply them with type tags. We’ll use the tag polynomial, and install
appropriate operations on tagged polynomials in the operation table.
We’ll embed all our code in an installation procedure for the polynomial
package, similar to the ones in Section 2.5.1:
(define (install-polynomial-package)
  ;; internal procedures
  ;; representation of poly
  (define (make-poly variable term-list) (cons variable term-list))
  (define (variable p) (car p))
  (define (term-list p) (cdr p))
  ⟨procedures same-variable? and variable? from section 2.3.2⟩
  ;; representation of terms and term lists
  ⟨procedures adjoin-term . . . coeff from text below ⟩
  (define (add-poly p1 p2) . . .)
  ⟨procedures used by add-poly⟩
  (define (mul-poly p1 p2) . . .)
  ⟨procedures used by mul-poly⟩
  ;; interface to rest of the system
  (define (tag p) (attach-tag 'polynomial p))
  (put 'add '(polynomial polynomial)
        (lambda (p1 p2) (tag (add-poly p1 p2))))
  (put 'mul '(polynomial polynomial)
        (lambda (p1 p2) (tag (mul-poly p1 p2))))
  (put 'make 'polynomial
        (lambda (var terms) (tag (make-poly var terms))))

Polynomial addition is performed termwise. Terms of the same order
(i.e., with the same power of the indeterminate) must be combined. is
is done by forming a new term of the same order whose coefficient is the
sum of the coefficients of the addends. Terms in one addend for which

there are no terms of the same order in the other addend are simply
accumulated into the sum polynomial being constructed.
    In order to manipulate term lists, we will assume that we have a
constructor the-empty-termlist that returns an empty term list and
a constructor adjoin-term that adjoins a new term to a term list. We
will also assume that we have a predicate empty-termlist? that tells if a
given term list is empty, a selector first-term that extracts the highest-
order term from a term list, and a selector rest-terms that returns all
but the highest-order term. To manipulate terms, we will suppose that
we have a constructor make-term that constructs a term with given or-
der and coefficient, and selectors order and coeff that return, respec-
tively, the order and the coefficient of the term. ese operations allow
us to consider both terms and term lists as data abstractions, whose
concrete representations we can worry about separately.
    Here is the procedure that constructs the term list for the sum of
two polynomials:56
(define (add-terms L1 L2)
  (cond ((empty-termlist? L1) L2)
           ((empty-termlist? L2) L1)
           (let ((t1 (first-term L1))
                   (t2 (first-term L2)))
              (cond ((> (order t1) (order t2))
                         t1 (add-terms (rest-terms L1) L2)))
                      ((< (order t1) (order t2))

  56 is   operation is very much like the ordered union-set operation we developed
in Exercise 2.62. In fact, if we think of the terms of the polynomial as a set ordered
according to the power of the indeterminate, then the program that produces the term
list for a sum is almost identical to union-set.

                      t2 (add-terms L1 (rest-terms L2))))
                      (make-term (order t1)
                                  (add (coeff t1) (coeff t2)))
                      (add-terms (rest-terms L1)
                                  (rest-terms L2)))))))))

e most important point to note here is that we used the generic ad-
dition procedure add to add together the coefficients of the terms being
combined. is has powerful consequences, as we will see below.
    In order to multiply two term lists, we multiply each term of the
first list by all the terms of the other list, repeatedly using mul-term-
by-all-terms, which multiplies a given term by all terms in a given
term list. e resulting term lists (one for each term of the first list) are
accumulated into a sum. Multiplying two terms forms a term whose
order is the sum of the orders of the factors and whose coefficient is the
product of the coefficients of the factors:
(define (mul-terms L1 L2)
  (if (empty-termlist? L1)
      (add-terms (mul-term-by-all-terms (first-term L1) L2)
                   (mul-terms (rest-terms L1) L2))))
(define (mul-term-by-all-terms t1 L)
  (if (empty-termlist? L)
      (let ((t2 (first-term L)))
          (make-term (+ (order t1) (order t2))
                       (mul (coeff t1) (coeff t2)))
          (mul-term-by-all-terms t1 (rest-terms L))))))

is is really all there is to polynomial addition and multiplication. No-
tice that, since we operate on terms using the generic procedures add
and mul, our polynomial package is automatically able to handle any
type of coefficient that is known about by the generic arithmetic pack-
age. If we include a coercion mechanism such as one of those discussed
in Section 2.5.2, then we also are automatically able to handle operations
on polynomials of different coefficient types, such as
                                      [                   ]
                    2                   4
                                            2 2
                [3x + (2 + 3i)x + 7] · x + x + (5 + 3i) .

Because we installed the polynomial addition and multiplication proce-
dures add-poly and mul-poly in the generic arithmetic system as the
add and mul operations for type polynomial, our system is also auto-
matically able to handle polynomial operations such as
         [                                  ] [                    ]
           (y + 1)x 2 + (y 2 + 1)x + (y − 1) · (y − 2)x + (y 3 + 7) .

e reason is that when the system tries to combine coefficients, it will
dispatch through add and mul. Since the coefficients are themselves
polynomials (in y), these will be combined using add-poly and mul-
poly. e result is a kind of “data-directed recursion” in which, for ex-
ample, a call to mul-poly will result in recursive calls to mul-poly in
order to multiply the coefficients. If the coefficients of the coefficients
were themselves polynomials (as might be used to represent polynomi-
als in three variables), the data direction would ensure that the system
would follow through another level of recursive calls, and so on through
as many levels as the structure of the data dictates.57
  57 Tomake this work completely smoothly, we should also add to our generic arith-
metic system the ability to coerce a “number” to a polynomial by regarding it as a

Representing term lists
Finally, we must confront the job of implementing a good representa-
tion for term lists. A term list is, in effect, a set of coefficients keyed
by the order of the term. Hence, any of the methods for representing
sets, as discussed in Section 2.3.3, can be applied to this task. On the
other hand, our procedures add-terms and mul-terms always access
term lists sequentially from highest to lowest order. us, we will use
some kind of ordered list representation.
    How should we structure the list that represents a term list? One
consideration is the “density” of the polynomials we intend to manip-
ulate. A polynomial is said to be dense if it has nonzero coefficients in
terms of most orders. If it has many zero terms it is said to be sparse. For
                      A : x 5 + 2x 4 + 3x 2 − 2x − 5
is a dense polynomial, whereas

                              B:      x 100 + 2x 2 + 1

is sparse.
    e term lists of dense polynomials are most efficiently represented
as lists of the coefficients. For example, A above would be nicely rep-
resented as (1 2 0 3 -2 -5). e order of a term in this representa-
tion is the length of the sublist beginning with that term’s coefficient,
polynomial of degree zero whose coefficient is the number. is is necessary if we are
going to perform operations such as

                         [x 2 + (y + 1)x + 5] + [x 2 + 2x + 1],

which requires adding the coefficient y + 1 to the coefficient 2.

decremented by 1.58 is would be a terrible representation for a sparse
polynomial such as B: ere would be a giant list of zeros punctuated
by a few lonely nonzero terms. A more reasonable representation of the
term list of a sparse polynomial is as a list of the nonzero terms, where
each term is a list containing the order of the term and the coefficient
for that order. In such a scheme, polynomial B is efficiently represented
as ((100 1) (2 2) (0 1)). As most polynomial manipulations are
performed on sparse polynomials, we will use this method. We will as-
sume that term lists are represented as lists of terms, arranged from
highest-order to lowest-order term. Once we have made this decision,
implementing the selectors and constructors for terms and term lists is
(define (adjoin-term term term-list)
  (if (=zero? (coeff term))
        (cons term term-list)))
(define (the-empty-termlist) '())
(define (first-term term-list) (car term-list))
(define (rest-terms term-list) (cdr term-list))
(define (empty-termlist? term-list) (null? term-list))
(define (make-term order coeff) (list order coeff))

   58 In these polynomial examples, we assume that we have implemented the generic

arithmetic system using the type mechanism suggested in Exercise 2.78. us, coeffi-
cients that are ordinary numbers will be represented as the numbers themselves rather
than as pairs whose car is the symbol scheme-number.
   59 Although we are assuming that term lists are ordered, we have implemented ad-

join-term to simply cons the new term onto the existing term list. We can get away
with this so long as we guarantee that the procedures (such as add-terms) that use ad-
join-term always call it with a higher-order term than appears in the list. If we did not
want to make such a guarantee, we could have implemented adjoin-term to be simi-
lar to the adjoin-set constructor for the ordered-list representation of sets (Exercise

(define (order term) (car term))
(define (coeff term) (cadr term))

where =zero? is as defined in Exercise 2.80. (See also Exercise 2.87 be-
   Users of the polynomial package will create (tagged) polynomials
by means of the procedure:
(define (make-polynomial var terms)
  ((get 'make 'polynomial) var terms))

      Exercise 2.87: Install =zero? for polynomials in the generic
      arithmetic package. is will allow adjoin-term to work
      for polynomials with coefficients that are themselves poly-

      Exercise 2.88: Extend the polynomial system to include
      subtraction of polynomials. (Hint: You may find it helpful
      to define a generic negation operation.)

      Exercise 2.89: Define procedures that implement the term-
      list representation described above as appropriate for dense

      Exercise 2.90: Suppose we want to have a polynomial sys-
      tem that is efficient for both sparse and dense polynomials.
      One way to do this is to allow both kinds of term-list repre-
      sentations in our system. e situation is analogous to the
      complex-number example of Section 2.4, where we allowed
      both rectangular and polar representations. To do this we
      must distinguish different types of term lists and make the
      operations on term lists generic. Redesign the polynomial

system to implement this generalization. is is a major ef-
fort, not a local change.

Exercise 2.91: A univariate polynomial can be divided by
another one to produce a polynomial quotient and a poly-
nomial remainder. For example,

             x5 − 1
                    = x 3 + x , remainder x − 1.
             x2 − 1
Division can be performed via long division. at is, divide
the highest-order term of the dividend by the highest-order
term of the divisor. e result is the first term of the quo-
tient. Next, multiply the result by the divisor, subtract that
from the dividend, and produce the rest of the answer by re-
cursively dividing the difference by the divisor. Stop when
the order of the divisor exceeds the order of the dividend
and declare the dividend to be the remainder. Also, if the
dividend ever becomes zero, return zero as both quotient
and remainder.
We can design a div-poly procedure on the model of add-
poly and mul-poly. e procedure checks to see if the two
polys have the same variable. If so, div-poly strips off the
variable and passes the problem to div-terms, which per-
forms the division operation on term lists. div-poly finally
reaaches the variable to the result supplied by div-terms.
It is convenient to design div-terms to compute both the
quotient and the remainder of a division. div-terms can
take two term lists as arguments and return a list of the
quotient term list and the remainder term list.

      Complete the following definition of div-terms by filling
      in the missing expressions. Use this to implement div-poly,
      which takes two polys as arguments and returns a list of the
      quotient and remainder polys.
      (define (div-terms L1 L2)
        (if (empty-termlist? L1)
            (list (the-empty-termlist) (the-empty-termlist))
            (let ((t1 (first-term L1))
                  (t2 (first-term L2)))
              (if (> (order t2) (order t1))
                  (list (the-empty-termlist) L1)
                  (let ((new-c (div (coeff t1) (coeff t2)))
                        (new-o (- (order t1) (order t2))))
                    (let ((rest-of-result
                           ⟨compute rest of result recursively ⟩ ))
                      ⟨form complete result⟩ ))))))

Hierarchies of types in symbolic algebra
Our polynomial system illustrates how objects of one type (polynomi-
als) may in fact be complex objects that have objects of many different
types as parts. is poses no real difficulty in defining generic opera-
tions. We need only install appropriate generic operations for perform-
ing the necessary manipulations of the parts of the compound types.
In fact, we saw that polynomials form a kind of “recursive data abstrac-
tion,” in that parts of a polynomial may themselves be polynomials. Our
generic operations and our data-directed programming style can handle
this complication without much trouble.
     On the other hand, polynomial algebra is a system for which the
data types cannot be naturally arranged in a tower. For instance, it is
possible to have polynomials in x whose coefficients are polynomials
in y. It is also possible to have polynomials in y whose coefficients are

polynomials in x. Neither of these types is “above” the other in any
natural way, yet it is oen necessary to add together elements from
each set. ere are several ways to do this. One possibility is to convert
one polynomial to the type of the other by expanding and rearrang-
ing terms so that both polynomials have the same principal variable.
One can impose a towerlike structure on this by ordering the variables
and thus always converting any polynomial to a “canonical form” with
the highest-priority variable dominant and the lower-priority variables
buried in the coefficients. is strategy works fairly well, except that
the conversion may expand a polynomial unnecessarily, making it hard
to read and perhaps less efficient to work with. e tower strategy is
certainly not natural for this domain or for any domain where the user
can invent new types dynamically using old types in various combining
forms, such as trigonometric functions, power series, and integrals.
    It should not be surprising that controlling coercion is a serious
problem in the design of large-scale algebraic-manipulation systems.
Much of the complexity of such systems is concerned with relationships
among diverse types. Indeed, it is fair to say that we do not yet com-
pletely understand coercion. In fact, we do not yet completely under-
stand the concept of a data type. Nevertheless, what we know provides
us with powerful structuring and modularity principles to support the
design of large systems.

      Exercise 2.92: By imposing an ordering on variables, ex-
      tend the polynomial package so that addition and multipli-
      cation of polynomials works for polynomials in different
      variables. (is is not easy!)

Extended exercise: Rational functions
We can extend our generic arithmetic system to include rational func-
tions. ese are “fractions” whose numerator and denominator are poly-
nomials, such as
                                x +1
                                x3 − 1
e system should be able to add, subtract, multiply, and divide rational
functions, and to perform such computations as

                 x +1     x      x 3 + 2x 2 + 3x + 1
                       +       =                     .
                 x3 − 1 x2 − 1    x4 + x3 − x − 1
(Here the sum has been simplified by removing common factors. Ordi-
nary “cross multiplication” would have produced a fourth-degree poly-
nomial over a fih-degree polynomial.)
   If we modify our rational-arithmetic package so that it uses generic
operations, then it will do what we want, except for the problem of
reducing fractions to lowest terms.

      Exercise 2.93: Modify the rational-arithmetic package to
      use generic operations, but change make-rat so that it does
      not aempt to reduce fractions to lowest terms. Test your
      system by calling make-rational on two polynomials to
      produce a rational function:
      (define p1 (make-polynomial 'x '((2 1) (0 1))))
      (define p2 (make-polynomial 'x '((3 1) (0 1))))
      (define rf (make-rational p2 p1))

      Now add rf to itself, using add. You will observe that this
      addition procedure does not reduce fractions to lowest terms.

We can reduce polynomial fractions to lowest terms using the same idea
we used with integers: modifying make-rat to divide both the numera-
tor and the denominator by their greatest common divisor. e notion
of “greatest common divisor” makes sense for polynomials. In fact, we
can compute the  of two polynomials using essentially the same
Euclid’s Algorithm that works for integers.60 e integer version is
(define (gcd a b)
  (if (= b 0)
        (gcd b (remainder a b))))

Using this, we could make the obvious modification to define a 
operation that works on term lists:
(define (gcd-terms a b)
  (if (empty-termlist? b)
        (gcd-terms b (remainder-terms a b))))

where remainder-terms picks out the remainder component of the list
returned by the term-list division operation div-terms that was imple-
mented in Exercise 2.91.

  60 e   fact that Euclid’s Algorithm works for polynomials is formalized in algebra
by saying that polynomials form a kind of algebraic domain called a Euclidean ring. A
Euclidean ring is a domain that admits addition, subtraction, and commutative mul-
tiplication, together with a way of assigning to each element x of the ring a positive
integer “measure” m(x ) with the properties that m(xy) ≥ m(x ) for any nonzero x and
y and that, given any x and y, there exists a q such that y = qx + r and either r = 0
or m(r ) < m(x ). From an abstract point of view, this is what is needed to prove that
Euclid’s Algorithm works. For the domain of integers, the measure m of an integer is
the absolute value of the integer itself. For the domain of polynomials, the measure of
a polynomial is its degree.

          Exercise 2.94: Using div-terms, implement the procedure
          remainder-terms and use this to define gcd-terms as above.
          Now write a procedure gcd-poly that computes the poly-
          nomial  of two polys. (e procedure should signal an
          error if the two polys are not in the same variable.) Install in
          the system a generic operation greatest-common-divisor
          that reduces to gcd-poly for polynomials and to ordinary
          gcd for ordinary numbers. As a test, try

          (define p1 (make-polynomial
                        'x '((4 1) (3 -1) (2 -2) (1 2))))
          (define p2 (make-polynomial 'x '((3 1) (1 -1))))
          (greatest-common-divisor p1 p2)

          and check your result by hand.

          Exercise 2.95: Define P1 , P2 , and P3 to be the polynomials

                                P1 : x 2 − 2x + 1,
                                P2 : 11x 2 + 7,
                                P3 : 13x + 5.

          Now define Q 1 to be the product of P1 and P2 and Q 2 to
          be the product of P1 and P3 , and use greatest-common-
          divisor (Exercise 2.94) to compute the  of Q 1 and Q 2 .
          Note that the answer is not the same as P1 . is example in-
          troduces noninteger operations into the computation, caus-
          ing difficulties with the  algorithm.61 To understand
  61 In  an implementation like  Scheme, this produces a polynomial that is indeed
a divisor of Q 1 and Q 2 , but with rational coefficients. In many other Scheme systems,
in which division of integers can produce limited-precision decimal numbers, we may
fail to get a valid divisor.

      what is happening, try tracing gcd-terms while comput-
      ing the  or try performing the division by hand.

We can solve the problem exhibited in Exercise 2.95 if we use the follow-
ing modification of the  algorithm (which really works only in the
case of polynomials with integer coefficients). Before performing any
polynomial division in the  computation, we multiply the dividend
by an integer constant factor, chosen to guarantee that no fractions will
arise during the division process. Our answer will thus differ from the
actual  by an integer constant factor, but this does not maer in the
case of reducing rational functions to lowest terms; the  will be used
to divide both the numerator and denominator, so the integer constant
factor will cancel out.
     More precisely, if P and Q are polynomials, let O 1 be the order of P
(i.e., the order of the largest term of P ) and let O 2 be the order of Q. Let c
be the leading coefficient of Q. en it can be shown that, if we multiply
P by the integerizing factor c 1+O 1 −O 2 , the resulting polynomial can be
divided by Q by using the div-terms algorithm without introducing any
fractions. e operation of multiplying the dividend by this constant
and then dividing is sometimes called the pseudodivision of P by Q. e
remainder of the division is called the pseudoremainder.

      Exercise 2.96:

         a. Implement the procedure pseudoremainder-terms, which
            is just like remainder-terms except that it multiplies
            the dividend by the integerizing factor described above
            before calling div-terms. Modify gcd-terms to use
            pseudoremainder-terms, and verify that greatest-
            common-divisor now produces an answer with inte-
            ger coefficients on the example in Exercise 2.95.
         b. e  now has integer coefficients, but they are
            larger than those of P1 . Modify gcd-terms so that it
            removes common factors from the coefficients of the
            answer by dividing all the coefficients by their (inte-
            ger) greatest common divisor.

us, here is how to reduce a rational function to lowest terms:

    • Compute the  of the numerator and denominator, using the
      version of gcd-terms from Exercise 2.96.

    • When you obtain the , multiply both numerator and denomi-
      nator by the same integerizing factor before dividing through by
      the , so that division by the  will not introduce any nonin-
      teger coefficients. As the factor you can use the leading coefficient
      of the  raised to the power 1 + O 1 − O 2 , where O 2 is the order
      of the  and O 1 is the maximum of the orders of the numerator
      and denominator. is will ensure that dividing the numerator
      and denominator by the  will not introduce any fractions.

    • e result of this operation will be a numerator and denominator
      with integer coefficients. e coefficients will normally be very
      large because of all of the integerizing factors, so the last step is to
      remove the redundant factors by computing the (integer) greatest
      common divisor of all the coefficients of the numerator and the
      denominator and dividing through by this factor.

      Exercise 2.97:

         a. Implement this algorithm as a procedure reduce-terms
            that takes two term lists n and d as arguments and re-

  turns a list nn, dd, which are n and d reduced to low-
  est terms via the algorithm given above. Also write a
  procedure reduce-poly, analogous to add-poly, that
  checks to see if the two polys have the same variable.
  If so, reduce-poly strips off the variable and passes
  the problem to reduce-terms, then reaaches the vari-
  able to the two term lists supplied by reduce-terms.
b. Define a procedure analogous to reduce-terms that
   does what the original make-rat did for integers:
  (define (reduce-integers n d)
    (let ((g (gcd n d)))
      (list (/ n g) (/ d g))))

  and define reduce as a generic operation that calls
  apply-generic to dispatch to either reduce-poly (for
  polynomial arguments) or reduce-integers (for scheme-
  number arguments). You can now easily make the rational-
  arithmetic package reduce fractions to lowest terms
  by having make-rat call reduce before combining the
  given numerator and denominator to form a ratio-
  nal number. e system now handles rational expres-
  sions in either integers or polynomials. To test your
  program, try the example at the beginning of this ex-
  tended exercise:
  (define   p1 (make-polynomial 'x '((1 1) (0     1))))
  (define   p2 (make-polynomial 'x '((3 1) (0 -1))))
  (define   p3 (make-polynomial 'x '((1 1))))
  (define   p4 (make-polynomial 'x '((2 1) (0 -1))))
  (define rf1 (make-rational p1 p2))
  (define rf2 (make-rational p3 p4))

              (add rf1 rf2)

              See if you get the correct answer, correctly reduced to
              lowest terms.

e  computation is at the heart of any system that does opera-
tions on rational functions. e algorithm used above, although mathe-
matically straightforward, is extremely slow. e slowness is due partly
to the large number of division operations and partly to the enormous
size of the intermediate coefficients generated by the pseudodivisions.
One of the active areas in the development of algebraic-manipulation
systems is the design of beer algorithms for computing polynomial

  62 One   extremely efficient and elegant method for computing polynomial s was
discovered by Richard Zippel (1979). e method is a probabilistic algorithm, as is the
fast test for primality that we discussed in Chapter 1. Zippel’s book (Zippel 1993) de-
scribes this method, together with other ways to compute polynomial s.

Modularity, Objects, and State

      Mεταβάλλον ὰναπαύεται
      (Even while it changes, it stands still.)

      Plus ça change, plus c’est la même chose.
      —Alphonse Karr

T      introduced the basic elements from which
    programs are made. We saw how primitive procedures and primi-
tive data are combined to construct compound entities, and we learned
that abstraction is vital in helping us to cope with the complexity of
large systems. But these tools are not sufficient for designing programs.
Effective program synthesis also requires organizational principles that
can guide us in formulating the overall design of a program. In partic-
ular, we need strategies to help us structure large systems so that they

will be modular, that is, so that they can be divided “naturally” into co-
herent parts that can be separately developed and maintained.
     One powerful design strategy, which is particularly appropriate to
the construction of programs for modeling physical systems, is to base
the structure of our programs on the structure of the system being mod-
eled. For each object in the system, we construct a corresponding com-
putational object. For each system action, we define a symbolic opera-
tion in our computational model. Our hope in using this strategy is that
extending the model to accommodate new objects or new actions will
require no strategic changes to the program, only the addition of the
new symbolic analogs of those objects or actions. If we have been suc-
cessful in our system organization, then to add a new feature or debug
an old one we will have to work on only a localized part of the system.
     To a large extent, then, the way we organize a large program is dic-
tated by our perception of the system to be modeled. In this chapter we
will investigate two prominent organizational strategies arising from
two rather different “world views” of the structure of systems. e first
organizational strategy concentrates on objects, viewing a large system
as a collection of distinct objects whose behaviors may change over
time. An alternative organizational strategy concentrates on the streams
of information that flow in the system, much as an electrical engineer
views a signal-processing system.
     Both the object-based approach and the stream-processing approach
raise significant linguistic issues in programming. With objects, we must
be concerned with how a computational object can change and yet main-
tain its identity. is will force us to abandon our old substitution model
of computation (Section 1.1.5) in favor of a more mechanistic but less
theoretically tractable environment model of computation. e difficul-
ties of dealing with objects, change, and identity are a fundamental con-

sequence of the need to grapple with time in our computational models.
ese difficulties become even greater when we allow the possibility of
concurrent execution of programs. e stream approach can be most
fully exploited when we decouple simulated time in our model from the
order of the events that take place in the computer during evaluation.
We will accomplish this using a technique known as delayed evaluation.

3.1 Assignment and Local State
We ordinarily view the world as populated by independent objects, each
of which has a state that changes over time. An object is said to “have
state” if its behavior is influenced by its history. A bank account, for
example, has state in that the answer to the question “Can I withdraw
$100?” depends upon the history of deposit and withdrawal transac-
tions. We can characterize an object’s state by one or more state vari-
ables, which among them maintain enough information about history
to determine the object’s current behavior. In a simple banking system,
we could characterize the state of an account by a current balance rather
than by remembering the entire history of account transactions.
    In a system composed of many objects, the objects are rarely com-
pletely independent. Each may influence the states of others through
interactions, which serve to couple the state variables of one object to
those of other objects. Indeed, the view that a system is composed of
separate objects is most useful when the state variables of the system
can be grouped into closely coupled subsystems that are only loosely
coupled to other subsystems.
    is view of a system can be a powerful framework for organizing
computational models of the system. For such a model to be modular, it
should be decomposed into computational objects that model the actual

objects in the system. Each computational object must have its own lo-
cal state variables describing the actual object’s state. Since the states of
objects in the system being modeled change over time, the state vari-
ables of the corresponding computational objects must also change. If
we choose to model the flow of time in the system by the elapsed time
in the computer, then we must have a way to construct computational
objects whose behaviors change as our programs run. In particular, if
we wish to model state variables by ordinary symbolic names in the
programming language, then the language must provide an assignment
operator to enable us to change the value associated with a name.

3.1.1 Local State Variables
To illustrate what we mean by having a computational object with time-
varying state, let us model the situation of withdrawing money from a
bank account. We will do this using a procedure withdraw, which takes
as argument an amount to be withdrawn. If there is enough money in
the account to accommodate the withdrawal, then withdraw should re-
turn the balance remaining aer the withdrawal. Otherwise, withdraw
should return the message Insufficient funds. For example, if we begin
with $100 in the account, we should obtain the following sequence of
responses using withdraw:
(withdraw 25)
(withdraw 25)
(withdraw 60)
"Insufficient funds"
(withdraw 15)

Observe that the expression (withdraw 25), evaluated twice, yields
different values. is is a new kind of behavior for a procedure. Until
now, all our procedures could be viewed as specifications for comput-
ing mathematical functions. A call to a procedure computed the value of
the function applied to the given arguments, and two calls to the same
procedure with the same arguments always produced the same result.1
     To implement withdraw, we can use a variable balance to indicate
the balance of money in the account and define withdraw as a procedure
that accesses balance. e withdraw procedure checks to see if balance
is at least as large as the requested amount. If so, withdraw decrements
balance by amount and returns the new value of balance. Otherwise,
withdraw returns the Insufficient funds message. Here are the definitions
of balance and withdraw:
(define balance 100)
(define (withdraw amount)
  (if (>= balance amount)
        (begin (set! balance (- balance amount))
        "Insufficient funds"))

Decrementing balance is accomplished by the expression
(set! balance (- balance amount))

is uses the set! special form, whose syntax is
(set!   ⟨name⟩ ⟨new-value⟩)
   1 Actually, this is not quite true. One exception was the random-number generator
in Section 1.2.6. Another exception involved the operation/type tables we introduced in
Section 2.4.3, where the values of two calls to get with the same arguments depended
on intervening calls to put. On the other hand, until we introduce assignment, we have
no way to create such procedures ourselves.

Here ⟨name ⟩ is a symbol and ⟨new-value ⟩ is any expression. set! changes
⟨name ⟩ so that its value is the result obtained by evaluating ⟨new-value ⟩.
In the case at hand, we are changing balance so that its new value will
be the result of subtracting amount from the previous value of balance.2
    withdraw also uses the begin special form to cause two expressions
to be evaluated in the case where the if test is true: first decrementing
balance and then returning the value of balance. In general, evaluating
the expression
(begin   ⟨exp1 ⟩ ⟨exp2 ⟩ . . . ⟨expk ⟩)

causes the expressions ⟨ exp1 ⟩ through ⟨ expk ⟩ to be evaluated in se-
quence and the value of the final expression ⟨ expk ⟩ to be returned as
the value of the entire begin form.3
    Although withdraw works as desired, the variable balance presents
a problem. As specified above, balance is a name defined in the global
environment and is freely accessible to be examined or modified by any
procedure. It would be much beer if we could somehow make balance
internal to withdraw, so that withdraw would be the only procedure
that could access balance directly and any other procedure could access
balance only indirectly (through calls to withdraw). is would more
accurately model the notion that balance is a local state variable used
   2 e value of a set! expression is implementation-dependent. set! should be used

only for its effect, not for its value.
  e name set! reflects a naming convention used in Scheme: Operations that change
the values of variables (or that change data structures, as we will see in Section 3.3) are
given names that end with an exclamation point. is is similar to the convention of
designating predicates by names that end with a question mark.
   3 We have already used begin implicitly in our programs, because in Scheme the

body of a procedure can be a sequence of expressions. Also, the ⟨consequent ⟩ part of
each clause in a cond expression can be a sequence of expressions rather than a single

by withdraw to keep track of the state of the account.
    We can make balance internal to withdraw by rewriting the defi-
nition as follows:
(define new-withdraw
  (let ((balance 100))
     (lambda (amount)
        (if (>= balance amount)
             (begin (set! balance (- balance amount))
             "Insufficient funds"))))

What we have done here is use let to establish an environment with a
local variable balance, bound to the initial value 100. Within this local
environment, we use lambda to create a procedure that takes amount as
an argument and behaves like our previous withdraw procedure. is
procedure—returned as the result of evaluating the let expression—is
new-withdraw, which behaves in precisely the same way as withdraw
but whose variable balance is not accessible by any other procedure.4
    Combining set! with local variables is the general programming
technique we will use for constructing computational objects with lo-
cal state. Unfortunately, using this technique raises a serious problem:
When we first introduced procedures, we also introduced the substi-
tution model of evaluation (Section 1.1.5) to provide an interpretation
of what procedure application means. We said that applying a proce-
dure should be interpreted as evaluating the body of the procedure with
the formal parameters replaced by their values. e trouble is that, as
   4 In programming-language jargon, the variable balance is said to be encapsulated

within the new-withdraw procedure. Encapsulation reflects the general system-design
principle known as the hiding principle: One can make a system more modular and ro-
bust by protecting parts of the system from each other; that is, by providing information
access only to those parts of the system that have a “need to know.”

soon as we introduce assignment into our language, substitution is no
longer an adequate model of procedure application. (We will see why
this is so in Section 3.1.3.) As a consequence, we technically have at
this point no way to understand why the new-withdraw procedure be-
haves as claimed above. In order to really understand a procedure such
as new-withdraw, we will need to develop a new model of procedure ap-
plication. In Section 3.2 we will introduce such a model, together with
an explanation of set! and local variables. First, however, we examine
some variations on the theme established by new-withdraw.
     e following procedure, make-withdraw, creates “withdrawal pro-
cessors.” e formal parameter balance in make-withdraw specifies the
initial amount of money in the account.5
(define (make-withdraw balance)
  (lambda (amount)
     (if (>= balance amount)
           (begin (set! balance (- balance amount))
           "Insufficient funds")))

make-withdraw       can be used as follows to create two objects W1 and W2:
(define W1 (make-withdraw 100))
(define W2 (make-withdraw 100))

(W1 50)
(W2 70)

    5 In contrast with new-withdraw above, we do not have to use let to make balance

a local variable, since formal parameters are already local. is will be clearer aer the
discussion of the environment model of evaluation in Section 3.2. (See also Exercise

(W2 40)
"Insufficient funds"
(W1 40)

Observe that W1 and W2 are completely independent objects, each with
its own local state variable balance. Withdrawals from one do not affect
the other.
     We can also create objects that handle deposits as well as with-
drawals, and thus we can represent simple bank accounts. Here is a
procedure that returns a “bank-account object” with a specified initial
(define (make-account balance)
  (define (withdraw amount)
     (if (>= balance amount)
          (begin (set! balance (- balance amount))
          "Insufficient funds"))
  (define (deposit amount)
     (set! balance (+ balance amount))
  (define (dispatch m)
     (cond ((eq? m 'withdraw) withdraw)
           ((eq? m 'deposit) deposit)
           (else (error "Unknown request: MAKE-ACCOUNT"

Each call to make-account sets up an environment with a local state
variable balance. Within this environment, make-account defines pro-
cedures deposit and withdraw that access balance and an additional
procedure dispatch that takes a “message” as input and returns one of

the two local procedures. e dispatch procedure itself is returned as
the value that represents the bank-account object. is is precisely the
message-passing style of programming that we saw in Section 2.4.3, al-
though here we are using it in conjunction with the ability to modify
local variables.
    make-account can be used as follows:

(define acc (make-account 100))
((acc 'withdraw) 50)
((acc 'withdraw) 60)
"Insufficient funds"
((acc 'deposit) 40)
((acc 'withdraw) 60)

Each call to acc returns the locally defined deposit or withdraw pro-
cedure, which is then applied to the specified amount. As was the case
with make-withdraw, another call to make-account
(define acc2 (make-account 100))

will produce a completely separate account object, which maintains its
own local balance.

      Exercise 3.1: An accumulator is a procedure that is called
      repeatedly with a single numeric argument and accumu-
      lates its arguments into a sum. Each time it is called, it
      returns the currently accumulated sum. Write a procedure
      make-accumulator that generates accumulators, each main-
      taining an independent sum. e input to make-accumulator
      should specify the initial value of the sum; for example

(define A (make-accumulator 5))
(A 10)
(A 10)

Exercise 3.2: In soware-testing applications, it is useful
to be able to count the number of times a given procedure
is called during the course of a computation. Write a pro-
cedure make-monitored that takes as input a procedure, f,
that itself takes one input. e result returned by make-
monitored is a third procedure, say mf, that keeps track
of the number of times it has been called by maintaining
an internal counter. If the input to mf is the special symbol
how-many-calls?, then mf returns the value of the counter.
If the input is the special symbol reset-count, then mf re-
sets the counter to zero. For any other input, mf returns the
result of calling f on that input and increments the counter.
For instance, we could make a monitored version of the
sqrt procedure:

(define s (make-monitored sqrt))
(s 100)
(s 'how-many-calls?)

Exercise 3.3: Modify the make-account procedure so that
it creates password-protected accounts. at is, make-account
should take a symbol as an additional argument, as in
(define acc (make-account 100 'secret-password))

      e resulting account object should process a request only
      if it is accompanied by the password with which the ac-
      count was created, and should otherwise return a complaint:
      ((acc 'secret-password 'withdraw) 40)
      ((acc 'some-other-password 'deposit) 50)
      "Incorrect password"

      Exercise 3.4: Modify the make-account procedure of Ex-
      ercise 3.3 by adding another local state variable so that, if
      an account is accessed more than seven consecutive times
      with an incorrect password, it invokes the procedure call-

3.1.2 The Benefits of Introducing Assignment
As we shall see, introducing assignment into our programming lan-
guage leads us into a thicket of difficult conceptual issues. Nevertheless,
viewing systems as collections of objects with local state is a powerful
technique for maintaining a modular design. As a simple example, con-
sider the design of a procedure rand that, whenever it is called, returns
an integer chosen at random.
    It is not at all clear what is meant by “chosen at random.” What we
presumably want is for successive calls to rand to produce a sequence of
numbers that has statistical properties of uniform distribution. We will
not discuss methods for generating suitable sequences here. Rather, let
us assume that we have a procedure rand-update that has the property
that if we start with a given number x 1 and form
x 2 = (rand-update x 1 )
x 3 = (rand-update x 2 )

then the sequence of values x 1 , x 2 , x 3 , . . . will have the desired statistical
    We can implement rand as a procedure with a local state variable
x that is initialized to some fixed value random-init. Each call to rand
computes rand-update of the current value of x, returns this as the
random number, and also stores this as the new value of x.
(define rand (let ((x random-init))
                   (lambda ()
                      (set! x (rand-update x))

Of course, we could generate the same sequence of random numbers
without using assignment by simply calling rand-update directly. How-
ever, this would mean that any part of our program that used random
numbers would have to explicitly remember the current value of x to
be passed as an argument to rand-update. To realize what an annoy-
ance this would be, consider using random numbers to implement a
technique called Monte Carlo simulation.
    e Monte Carlo method consists of choosing sample experiments
at random from a large set and then making deductions on the basis of
    6 One common way to implement rand-update is to use the rule that x is updated

to ax + b modulo m, where a, b, and m are appropriately chosen integers. Chapter 3
of Knuth 1981 includes an extensive discussion of techniques for generating sequences
of random numbers and establishing their statistical properties. Notice that the rand-
update procedure computes a mathematical function: Given the same input twice, it
produces the same output. erefore, the number sequence produced by rand-update
certainly is not “random,” if by “random” we insist that each number in the sequence
is unrelated to the preceding number. e relation between “real randomness” and so-
called pseudo-random sequences, which are produced by well-determined computations
and yet have suitable statistical properties, is a complex question involving difficult
issues in mathematics and philosophy. Kolmogorov, Solomonoff, and Chaitin have made
great progress in clarifying these issues; a discussion can be found in Chaitin 1975.

the probabilities estimated from tabulating the results of those experi-
ments. For example, we can approximate π using the fact that 6/π 2 is
the probability that two integers chosen at random will have no fac-
tors in common; that is, that their greatest common divisor will be 1.7
To obtain the approximation to π , we perform a large number of ex-
periments. In each experiment we choose two integers at random and
perform a test to see if their  is 1. e fraction of times that the test
is passed gives us our estimate of 6/π 2 , and from this we obtain our
approximation to π .
    e heart of our program is a procedure monte-carlo, which takes
as arguments the number of times to try an experiment, together with
the experiment, represented as a no-argument procedure that will re-
turn either true or false each time it is run. monte-carlo runs the exper-
iment for the designated number of trials and returns a number telling
the fraction of the trials in which the experiment was found to be true.
(define (estimate-pi trials)
  (sqrt (/ 6 (monte-carlo trials cesaro-test))))
(define (cesaro-test)
   (= (gcd (rand) (rand)) 1))

(define (monte-carlo trials experiment)
  (define (iter trials-remaining trials-passed)
     (cond ((= trials-remaining 0)
              (/ trials-passed trials))
              (iter (- trials-remaining 1)
                      (+ trials-passed 1)))

   7 is theorem is due to E. Cesàro. See section 4.5.2 of Knuth 1981 for a discussion

and a proof.

           (iter (- trials-remaining 1)
  (iter trials 0))

Now let us try the same computation using rand-update directly rather
than rand, the way we would be forced to proceed if we did not use
assignment to model local state:
(define (estimate-pi trials)
  (sqrt (/ 6 (random-gcd-test trials random-init))))
(define (random-gcd-test trials initial-x)
  (define (iter trials-remaining trials-passed x)
    (let ((x1 (rand-update x)))
      (let ((x2 (rand-update x1)))
        (cond ((= trials-remaining 0)
                (/ trials-passed trials))
              ((= (gcd x1 x2) 1)
                (iter (- trials-remaining 1)
                      (+ trials-passed 1)
                (iter (- trials-remaining 1)
  (iter trials 0 initial-x))

While the program is still simple, it betrays some painful breaches of
modularity. In our first version of the program, using rand, we can ex-
press the Monte Carlo method directly as a general monte-carlo proce-
dure that takes as an argument an arbitrary experiment procedure. In
our second version of the program, with no local state for the random-
number generator, random-gcd-test must explicitly manipulate the ran-
dom numbers x1 and x2 and recycle x2 through the iterative loop as
the new input to rand-update. is explicit handling of the random

numbers intertwines the structure of accumulating test results with the
fact that our particular experiment uses two random numbers, whereas
other Monte Carlo experiments might use one random number or three.
Even the top-level procedure estimate-pi has to be concerned with
supplying an initial random number. e fact that the random-number
generator’s insides are leaking out into other parts of the program makes
it difficult for us to isolate the Monte Carlo idea so that it can be applied
to other tasks. In the first version of the program, assignment encapsu-
lates the state of the random-number generator within the rand proce-
dure, so that the details of random-number generation remain indepen-
dent of the rest of the program.
     e general phenomenon illustrated by the Monte Carlo example is
this: From the point of view of one part of a complex process, the other
parts appear to change with time. ey have hidden time-varying local
state. If we wish to write computer programs whose structure reflects
this decomposition, we make computational objects (such as bank ac-
counts and random-number generators) whose behavior changes with
time. We model state with local state variables, and we model the changes
of state with assignments to those variables.
     It is tempting to conclude this discussion by saying that, by intro-
ducing assignment and the technique of hiding state in local variables,
we are able to structure systems in a more modular fashion than if all
state had to be manipulated explicitly, by passing additional parameters.
Unfortunately, as we shall see, the story is not so simple.

      Exercise 3.5: Monte Carlo integration is a method of esti-
      mating definite integrals by means of Monte Carlo simula-
      tion. Consider computing the area of a region of space de-
      scribed by a predicate P (x , y) that is true for points (x , y)
      in the region and false for points not in the region. For

example, the region contained within a circle of radius 3
centered at (5, 7) is described by the predicate that tests
whether (x − 5)2 + (y − 7)2 ≤ 32 . To estimate the area of the
region described by such a predicate, begin by choosing a
rectangle that contains the region. For example, a rectangle
with diagonally opposite corners at (2, 4) and (8, 10) con-
tains the circle above. e desired integral is the area of
that portion of the rectangle that lies in the region. We can
estimate the integral by picking, at random, points (x , y)
that lie in the rectangle, and testing P (x , y) for each point
to determine whether the point lies in the region. If we try
this with many points, then the fraction of points that fall
in the region should give an estimate of the proportion of
the rectangle that lies in the region. Hence, multiplying this
fraction by the area of the entire rectangle should produce
an estimate of the integral.
Implement Monte Carlo integration as a procedure estimate-
integral that takes as arguments a predicate P, upper and
lower bounds x1, x2, y1, and y2 for the rectangle, and the
number of trials to perform in order to produce the esti-
mate. Your procedure should use the same monte-carlo
procedure that was used above to estimate π . Use your estimate-
integral to produce an estimate of π by measuring the
area of a unit circle.
You will find it useful to have a procedure that returns a
number chosen at random from a given range. e follow-
ing random-in-range procedure implements this in terms
of the random procedure used in Section 1.2.6, which re-

       turns a nonnegative number less than its input.8
       (define (random-in-range low high)
          (let ((range (- high low)))
             (+ low (random range))))

       Exercise 3.6: It is useful to be able to reset a random-number
       generator to produce a sequence starting from a given value.
       Design a new rand procedure that is called with an ar-
       gument that is either the symbol generate or the symbol
       reset and behaves as follows: (rand 'generate) produces
       a new random number; ((rand 'reset) ⟨new-value ⟩) re-
       sets the internal state variable to the designated ⟨new-value ⟩.
       us, by reseing the state, one can generate repeatable se-
       quences. ese are very handy to have when testing and
       debugging programs that use random numbers.

3.1.3 The Costs of Introducing Assignment
As we have seen, the set! operation enables us to model objects that
have local state. However, this advantage comes at a price. Our pro-
gramming language can no longer be interpreted in terms of the sub-
stitution model of procedure application that we introduced in Section
1.1.5. Moreover, no simple model with “nice” mathematical properties
can be an adequate framework for dealing with objects and assignment
in programming languages.
    So long as we do not use assignments, two evaluations of the same
procedure with the same arguments will produce the same result, so
   8  Scheme provides such a procedure. If random is given an exact integer (as in

Section 1.2.6) it returns an exact integer, but if it is given a decimal value (as in this
exercise) it returns a decimal value.

that procedures can be viewed as computing mathematical functions.
Programming without any use of assignments, as we did throughout
the first two chapters of this book, is accordingly known as functional
    To understand how assignment complicates maers, consider a sim-
plified version of the make-withdraw procedure of Section 3.1.1 that
does not bother to check for an insufficient amount:
(define (make-simplified-withdraw balance)
    (lambda (amount)
      (set! balance (- balance amount))
(define W (make-simplified-withdraw 25))
(W 20)
(W 10)

Compare this procedure with the following make-decrementer proce-
dure, which does not use set!:
(define (make-decrementer balance)
    (lambda (amount)
      (- balance amount)))

make-decrementer     returns a procedure that subtracts its input from a
designated amount balance, but there is no accumulated effect over
successive calls, as with make-simplified-withdraw:
(define D (make-decrementer 25))
(D 20)
(D 10)

We can use the substitution model to explain how make-decrementer
works. For instance, let us analyze the evaluation of the expression
((make-decrementer 25) 20)

We first simplify the operator of the combination by substituting 25 for
balance in the body of make-decrementer. is reduces the expression
((lambda (amount) (- 25 amount)) 20)

Now we apply the operator by substituting 20 for amount in the body
of the lambda expression:
(- 25 20)

e final answer is 5.
   Observe, however, what happens if we aempt a similar substitution
analysis with make-simplified-withdraw:
((make-simplified-withdraw 25) 20)

We first simplify the operator by substituting 25 for balance in the body
of make-simplified-withdraw. is reduces the expression to9
((lambda (amount) (set! balance (- 25 amount)) 25) 20)

Now we apply the operator by substituting 20 for amount in the body
of the lambda expression:
(set! balance (- 25 20)) 25

If we adhered to the substitution model, we would have to say that the
meaning of the procedure application is to first set balance to 5 and then
   9 We don’t substitute for the occurrence of balance in the set! expression because

the ⟨name ⟩ in a set! is not evaluated. If we did substitute for it, we would get (set!
25 (- 25 amount)), which makes no sense.

return 25 as the value of the expression. is gets the wrong answer. In
order to get the correct answer, we would have to somehow distinguish
the first occurrence of balance (before the effect of the set!) from the
second occurrence of balance (aer the effect of the set!), and the
substitution model cannot do this.
    e trouble here is that substitution is based ultimately on the no-
tion that the symbols in our language are essentially names for values.
But as soon as we introduce set! and the idea that the value of a vari-
able can change, a variable can no longer be simply a name. Now a
variable somehow refers to a place where a value can be stored, and the
value stored at this place can change. In Section 3.2 we will see how
environments play this role of “place” in our computational model.

Sameness and change
e issue surfacing here is more profound than the mere breakdown of
a particular model of computation. As soon as we introduce change into
our computational models, many notions that were previously straight-
forward become problematical. Consider the concept of two things be-
ing “the same.”
    Suppose we call make-decrementer twice with the same argument
to create two procedures:
(define D1 (make-decrementer 25))
(define D2 (make-decrementer 25))

Are D1 and D2 the same? An acceptable answer is yes, because D1 and D2
have the same computational behavior—each is a procedure that sub-
tracts its input from 25. In fact, D1 could be substituted for D2 in any
computation without changing the result.
    Contrast this with making two calls to make-simplified-withdraw:

(define W1 (make-simplified-withdraw 25))
(define W2 (make-simplified-withdraw 25))

Are W1 and W2 the same? Surely not, because calls to W1 and W2 have
distinct effects, as shown by the following sequence of interactions:
(W1 20)
(W1 20)
(W2 20)

Even though W1 and W2 are “equal” in the sense that they are both cre-
ated by evaluating the same expression, (make-simplified-withdraw
25), it is not true that W1 could be substituted for W2 in any expression
without changing the result of evaluating the expression.
     A language that supports the concept that “equals can be substituted
for equals” in an expression without changing the value of the expres-
sion is said to be referentially transparent. Referential transparency is
violated when we include set! in our computer language. is makes
it tricky to determine when we can simplify expressions by substituting
equivalent expressions. Consequently, reasoning about programs that
use assignment becomes drastically more difficult.
     Once we forgo referential transparency, the notion of what it means
for computational objects to be “the same” becomes difficult to capture
in a formal way. Indeed, the meaning of “same” in the real world that our
programs model is hardly clear in itself. In general, we can determine
that two apparently identical objects are indeed “the same one” only by
modifying one object and then observing whether the other object has
changed in the same way. But how can we tell if an object has “changed”
other than by observing the “same” object twice and seeing whether

some property of the object differs from one observation to the next?
us, we cannot determine “change” without some a priori notion of
“sameness,” and we cannot determine sameness without observing the
effects of change.
    As an example of how this issue arises in programming, consider
the situation where Peter and Paul have a bank account with $100 in it.
ere is a substantial difference between modeling this as
(define peter-acc (make-account 100))
(define paul-acc (make-account 100))

and modeling it as
(define peter-acc (make-account 100))
(define paul-acc peter-acc)

In the first situation, the two bank accounts are distinct. Transactions
made by Peter will not affect Paul’s account, and vice versa. In the sec-
ond situation, however, we have defined paul-acc to be the same thing
as peter-acc. In effect, Peter and Paul now have a joint bank account,
and if Peter makes a withdrawal from peter-acc Paul will observe less
money in paul-acc. ese two similar but distinct situations can cause
confusion in building computational models. With the shared account,
in particular, it can be especially confusing that there is one object (the
bank account) that has two different names (peter-acc and paul-acc);
if we are searching for all the places in our program where paul-acc
can be changed, we must remember to look also at things that change
  10 e phenomenon of a single computational object being accessed by more than one

name is known as aliasing. e joint bank account situation illustrates a very simple
example of an alias. In Section 3.3 we will see much more complex examples, such as
“distinct” compound data structures that share parts. Bugs can occur in our programs

     With reference to the above remarks on “sameness” and “change,”
observe that if Peter and Paul could only examine their bank balances,
and could not perform operations that changed the balance, then the is-
sue of whether the two accounts are distinct would be moot. In general,
so long as we never modify data objects, we can regard a compound
data object to be precisely the totality of its pieces. For example, a ratio-
nal number is determined by giving its numerator and its denominator.
But this view is no longer valid in the presence of change, where a com-
pound data object has an “identity” that is something different from the
pieces of which it is composed. A bank account is still “the same” bank
account even if we change the balance by making a withdrawal; con-
versely, we could have two different bank accounts with the same state
information. is complication is a consequence, not of our program-
ming language, but of our perception of a bank account as an object. We
do not, for example, ordinarily regard a rational number as a change-
able object with identity, such that we could change the numerator and
still have “the same” rational number.

Pitfalls of imperative programming
In contrast to functional programming, programming that makes ex-
tensive use of assignment is known as imperative programming. In ad-
dition to raising complications about computational models, programs
wrien in imperative style are susceptible to bugs that cannot occur in
functional programs. For example, recall the iterative factorial program
if we forget that a change to an object may also, as a “side effect,” change a “different”
object because the two “different” objects are actually a single object appearing under
different aliases. ese so-called side-effect bugs are so difficult to locate and to analyze
that some people have proposed that programming languages be designed in such a
way as to not allow side effects or aliasing (Lampson et al. 1981; Morris et al. 1980).

from Section 1.2.1:
(define (factorial n)
  (define (iter product counter)
    (if (> counter n)
        (iter (* counter product) (+ counter 1))))
  (iter 1 1))

Instead of passing arguments in the internal iterative loop, we could
adopt a more imperative style by using explicit assignment to update
the values of the variables product and counter:
(define (factorial n)
  (let ((product 1)
        (counter 1))
    (define (iter)
      (if (> counter n)
           (begin (set! product (* counter product))
                      (set! counter (+ counter 1))

is does not change the results produced by the program, but it does
introduce a subtle trap. How do we decide the order of the assignments?
As it happens, the program is correct as wrien. But writing the assign-
ments in the opposite order
(set! counter (+ counter 1))
(set! product (* counter product))

would have produced a different, incorrect result. In general, program-
ming with assignment forces us to carefully consider the relative orders
of the assignments to make sure that each statement is using the correct

version of the variables that have been changed. is issue simply does
not arise in functional programs.11
    e complexity of imperative programs becomes even worse if we
consider applications in which several processes execute concurrently.
We will return to this in Section 3.4. First, however, we will address the
issue of providing a computational model for expressions that involve
assignment, and explore the uses of objects with local state in designing

          Exercise 3.7: Consider the bank account objects created by
          make-account, with the password modification described
          in Exercise 3.3. Suppose that our banking system requires
          the ability to make joint accounts. Define a procedure make-
          joint that accomplishes this. make-joint should take three
          arguments. e first is a password-protected account. e
          second argument must match the password with which the
          account was defined in order for the make-joint operation
          to proceed. e third argument is a new password. make-
          joint is to create an additional access to the original ac-
          count using the new password. For example, if peter-acc
          is a bank account with password open-sesame, then
          (define paul-acc
            (make-joint peter-acc 'open-sesame 'rosebud))

  11 In view of this, it is ironic that introductory programming is most oen taught
in a highly imperative style. is may be a vestige of a belief, common throughout
the 1960s and 1970s, that programs that call procedures must inherently be less effi-
cient than programs that perform assignments. (Steele 1977 debunks this argument.)
Alternatively it may reflect a view that step-by-step assignment is easier for beginners
to visualize than procedure call. Whatever the reason, it oen saddles beginning pro-
grammers with “should I set this variable before or aer that one” concerns that can
complicate programming and obscure the important ideas.

      will allow one to make transactions on peter-acc using the
      name paul-acc and the password rosebud. You may wish
      to modify your solution to Exercise 3.3 to accommodate this
      new feature.

      Exercise 3.8: When we defined the evaluation model in
      Section 1.1.3, we said that the first step in evaluating an
      expression is to evaluate its subexpressions. But we never
      specified the order in which the subexpressions should be
      evaluated (e.g., le to right or right to le). When we in-
      troduce assignment, the order in which the arguments to a
      procedure are evaluated can make a difference to the result.
      Define a simple procedure f such that evaluating
      (+ (f 0) (f 1))

      will return 0 if the arguments to + are evaluated from le to
      right but will return 1 if the arguments are evaluated from
      right to le.

3.2 The Environment Model of Evaluation
When we introduced compound procedures in Chapter 1, we used the
substitution model of evaluation (Section 1.1.5) to define what is meant
by applying a procedure to arguments:

    • To apply a compound procedure to arguments, evaluate the body
      of the procedure with each formal parameter replaced by the cor-
      responding argument.

Once we admit assignment into our programming language, such a def-
inition is no longer adequate. In particular, Section 3.1.3 argued that, in

the presence of assignment, a variable can no longer be considered to be
merely a name for a value. Rather, a variable must somehow designate
a “place” in which values can be stored. In our new model of evaluation,
these places will be maintained in structures called environments.
    An environment is a sequence of frames. Each frame is a table (pos-
sibly empty) of bindings, which associate variable names with their cor-
responding values. (A single frame may contain at most one binding
for any variable.) Each frame also has a pointer to its enclosing environ-
ment, unless, for the purposes of discussion, the frame is considered to
be global. e value of a variable with respect to an environment is the
value given by the binding of the variable in the first frame in the en-
vironment that contains a binding for that variable. If no frame in the
sequence specifies a binding for the variable, then the variable is said to
be unbound in the environment.
    Figure 3.1 shows a simple environment structure consisting of three
frames, labeled I, II, and III. In the diagram, A, B, C, and D are pointers to
environments. C and D point to the same environment. e variables z
and x are bound in frame II, while y and x are bound in frame I. e value
of x in environment D is 3. e value of x with respect to environment
B is also 3. is is determined as follows: We examine the first frame in
the sequence (frame III) and do not find a binding for x, so we proceed
to the enclosing environment D and find the binding in frame I. On the
other hand, the value of x in environment A is 7, because the first frame
in the sequence (frame II) contains a binding of x to 7. With respect to
environment A, the binding of x to 7 in frame II is said to shadow the
binding of x to 3 in frame I.
    e environment is crucial to the evaluation process, because it de-
termines the context in which an expression should be evaluated. In-
deed, one could say that expressions in a programming language do


                                  C         D

                             II                       III
                      z:6                       m:1
                      x:7                       y:2

                         A                        B

             Figure 3.1: A simple environment structure.

not, in themselves, have any meaning. Rather, an expression acquires a
meaning only with respect to some environment in which it is evalu-
ated. Even the interpretation of an expression as straightforward as (+
1 1) depends on an understanding that one is operating in a context in
which + is the symbol for addition. us, in our model of evaluation we
will always speak of evaluating an expression with respect to some envi-
ronment. To describe interactions with the interpreter, we will suppose
that there is a global environment, consisting of a single frame (with no
enclosing environment) that includes values for the symbols associated
with the primitive procedures. For example, the idea that + is the sym-
bol for addition is captured by saying that the symbol + is bound in the
global environment to the primitive addition procedure.

3.2.1 The Rules for Evaluation
e overall specification of how the interpreter evaluates a combination
remains the same as when we first introduced it in Section 1.1.3:

     • To evaluate a combination:

    1. Evaluate the subexpressions of the combination.12

    2. Apply the value of the operator subexpression to the values of the
       operand subexpressions.

e environment model of evaluation replaces the substitution model in
specifying what it means to apply a compound procedure to arguments.
    In the environment model of evaluation, a procedure is always a pair
consisting of some code and a pointer to an environment. Procedures
are created in one way only: by evaluating a λ-expression. is produces
a procedure whose code is obtained from the text of the λ-expression
and whose environment is the environment in which the λ-expression
was evaluated to produce the procedure. For example, consider the pro-
cedure definition
(define (square x)
  (* x x))

evaluated in the global environment. e procedure definition syntax
is just syntactic sugar for an underlying implicit λ-expression. It would
have been equivalent to have used
(define square
  (lambda (x) (* x x)))

  12 Assignment    introduces a subtlety into step 1 of the evaluation rule. As shown in
Exercise 3.8, the presence of assignment allows us to write expressions that will produce
different values depending on the order in which the subexpressions in a combination
are evaluated. us, to be precise, we should specify an evaluation order in step 1 (e.g.,
le to right or right to le). However, this order should always be considered to be
an implementation detail, and one should never write programs that depend on some
particular order. For instance, a sophisticated compiler might optimize a program by
varying the order in which subexpressions are evaluated.

                    global      other variables
                    env         square:

                   (define (square x)
                     (* x x))

                                   parameters: x
                                   body: (* x x)

      Figure 3.2: Environment structure produced by evaluating
      (define (square x) (* x x)) in the global environment.

which evaluates (lambda (x) (* x x)) and binds square to the re-
sulting value, all in the global environment.
    Figure 3.2 shows the result of evaluating this define expression.
e procedure object is a pair whose code specifies that the procedure
has one formal parameter, namely x, and a procedure body (* x x).
e environment part of the procedure is a pointer to the global envi-
ronment, since that is the environment in which the λ-expression was
evaluated to produce the procedure. A new binding, which associates
the procedure object with the symbol square, has been added to the
global frame. In general, define creates definitions by adding bindings
to frames.
    Now that we have seen how procedures are created, we can describe
how procedures are applied. e environment model specifies: To ap-
ply a procedure to arguments, create a new environment containing a
frame that binds the parameters to the values of the arguments. e en-
closing environment of this frame is the environment specified by the

           global        other variables
           env           square:

          (square 5)
                                              E1       x:5

                                                      (* x x)
                            parameters: x
                            body: (* x x)

      Figure 3.3: Environment created by evaluating (square 5)
      in the global environment.

procedure. Now, within this new environment, evaluate the procedure
     To show how this rule is followed, Figure 3.3 illustrates the environ-
ment structure created by evaluating the expression (square 5) in the
global environment, where square is the procedure generated in Figure
3.2. Applying the procedure results in the creation of a new environ-
ment, labeled E1 in the figure, that begins with a frame in which x, the
formal parameter for the procedure, is bound to the argument 5. e
pointer leading upward from this frame shows that the frame’s enclos-
ing environment is the global environment. e global environment is
chosen here, because this is the environment that is indicated as part
of the square procedure object. Within E1, we evaluate the body of the
procedure, (* x x). Since the value of x in E1 is 5, the result is (* 5
5), or 25.
     e environment model of procedure application can be summa-
rized by two rules:

     • A procedure object is applied to a set of arguments by construct-
       ing a frame, binding the formal parameters of the procedure to the
       arguments of the call, and then evaluating the body of the proce-
       dure in the context of the new environment constructed. e new
       frame has as its enclosing environment the environment part of
       the procedure object being applied.

     • A procedure is created by evaluating a λ-expression relative to a
       given environment. e resulting procedure object is a pair con-
       sisting of the text of the λ-expression and a pointer to the envi-
       ronment in which the procedure was created.

We also specify that defining a symbol using define creates a binding
in the current environment frame and assigns to the symbol the indi-
cated value.13 Finally, we specify the behavior of set!, the operation
that forced us to introduce the environment model in the first place.
Evaluating the expression (set! ⟨variable ⟩ ⟨value ⟩) in some environ-
ment locates the binding of the variable in the environment and changes
that binding to indicate the new value. at is, one finds the first frame
in the environment that contains a binding for the variable and modi-
fies that frame. If the variable is unbound in the environment, then set!
signals an error.
    ese evaluation rules, though considerably more complex than the
substitution model, are still reasonably straightforward. Moreover, the
evaluation model, though abstract, provides a correct description of
  13 If there is already a binding for the variable in the current frame, then the binding is

changed. is is convenient because it allows redefinition of symbols; however, it also
means that define can be used to change values, and this brings up the issues of assign-
ment without explicitly using set!. Because of this, some people prefer redefinitions
of existing symbols to signal errors or warnings.

how the interpreter evaluates expressions. In Chapter 4 we shall see
how this model can serve as a blueprint for implementing a working
interpreter. e following sections elaborate the details of the model by
analyzing some illustrative programs.

3.2.2 Applying Simple Procedures
When we introduced the substitution model in Section 1.1.5 we showed
how the combination (f 5) evaluates to 136, given the following pro-
cedure definitions:
(define (square x)
  (* x x))
(define (sum-of-squares x y)
  (+ (square x) (square y)))
(define (f a)
  (sum-of-squares (+ a 1) (* a 2)))

We can analyze the same example using the environment model. Figure
3.4 shows the three procedure objects created by evaluating the defini-
tions of f, square, and sum-of-squares in the global environment. Each
procedure object consists of some code, together with a pointer to the
global environment.
    In Figure 3.5 we see the environment structure created by evaluat-
ing the expression (f 5). e call to f creates a new environment E1
beginning with a frame in which a, the formal parameter of f, is bound
to the argument 5. In E1, we evaluate the body of f:
(sum-of-squares (+ a 1) (* a 2))

To evaluate this combination, we first evaluate the subexpressions. e
first subexpression, sum-of-squares, has a value that is a procedure ob-
ject. (Notice how this value is found: We first look in the first frame of

 global        square:
 env           f:

      parameters: a              parameters: x   parameters: x, y
      body: (sum-of-squares      body: (* x x)   body: (+ (square x)
              (+ a 1)                                    (square y))
              (* a 2))

          Figure 3.4: Procedure objects in the global frame.

E1, which contains no binding for sum-of-squares. en we proceed
to the enclosing environment, i.e. the global environment, and find the
binding shown in Figure 3.4.) e other two subexpressions are evalu-
ated by applying the primitive operations + and * to evaluate the two
combinations (+ a 1) and (* a 2) to obtain 6 and 10, respectively.
     Now we apply the procedure object sum-of-squares to the argu-
ments 6 and 10. is results in a new environment E2 in which the
formal parameters x and y are bound to the arguments. Within E2 we
evaluate the combination (+ (square x) (square y)). is leads us to
evaluate (square x), where square is found in the global frame and x
is 6. Once again, we set up a new environment, E3, in which x is bound
to 6, and within this we evaluate the body of square, which is (* x x).
Also as part of applying sum-of-squares, we must evaluate the subex-
pression (square y), where y is 10. is second call to square creates
another environment, E4, in which x, the formal parameter of square,
is bound to 10. And within E4 we must evaluate (* x x).


   (f 5)

              a:5             x:6             x:6             x:10
        E1             E2               E3              E4

     (sum-of-squares     (+ (square x)       (* x x)         (* x x)
       (+ a 1)              (square y))
       (* a 2))

      Figure 3.5: Environments created by evaluating (f 5) us-
      ing the procedures in Figure 3.4.

    e important point to observe is that each call to square creates a
new environment containing a binding for x. We can see here how the
different frames serve to keep separate the different local variables all
named x. Notice that each frame created by square points to the global
environment, since this is the environment indicated by the square pro-
cedure object.
    Aer the subexpressions are evaluated, the results are returned. e
values generated by the two calls to square are added by sum-of-squares,
and this result is returned by f. Since our focus here is on the environ-
ment structures, we will not dwell on how these returned values are
passed from call to call; however, this is also an important aspect of the
evaluation process, and we will return to it in detail in Chapter 5.

      Exercise 3.9: In Section 1.2.1 we used the substitution model
      to analyze two procedures for computing factorials, a recur-
      sive version

       (define (factorial n)
          (if (= n 1) 1 (* n (factorial (- n 1)))))

       and an iterative version
       (define (factorial n) (fact-iter 1 1 n))
       (define (fact-iter product counter max-count)
          (if (> counter max-count)
               (fact-iter (* counter product)
                              (+ counter 1)

       Show the environment structures created by evaluating
       (factorial 6) using each version of the factorial pro-

3.2.3 Frames as the Repository of Local State
We can turn to the environment model to see how procedures and as-
signment can be used to represent objects with local state. As an exam-
ple, consider the “withdrawal processor” from Section 3.1.1 created by
calling the procedure
(define (make-withdraw balance)
  (lambda (amount)
     (if (>= balance amount)
          (begin (set! balance (- balance amount))
          "Insufficient funds")))

  14 e   environment model will not clarify our claim in Section 1.2.1 that the inter-
preter can execute a procedure such as fact-iter in a constant amount of space using
tail recursion. We will discuss tail recursion when we deal with the control structure
of the interpreter in Section 5.4.


          parameters: balance
          body: (lambda (amount)
                  (if (>= balance amount)
                      (begin (set! balance (-- balance amount))
                      "insufficient funds"))

      Figure 3.6: Result of defining make-withdraw in the global

Let us describe the evaluation of
(define W1 (make-withdraw 100))

followed by
(W1 50)

Figure 3.6 shows the result of defining the make-withdraw procedure in
the global environment. is produces a procedure object that contains
a pointer to the global environment. So far, this is no different from the
examples we have already seen, except that the body of the procedure
is itself a λ-expression.
     e interesting part of the computation happens when we apply the
procedure make-withdraw to an argument:
(define W1 (make-withdraw 100))

     global      make-withdraw:
     env         W1:

                          E1      balance: 100

                                                 parameters: balance
                                                 body: ...
     parameters: amount
     body: (if (>= balance amount)
               (begin (set! balance (- balance amount))
               "insufficient funds")

Figure 3.7: Result of evaluating (define W1 (make-withdraw 100)).

We begin, as usual, by seing up an environment E1 in which the formal
parameter balance is bound to the argument 100. Within this environ-
ment, we evaluate the body of make-withdraw, namely the λ-expression.
is constructs a new procedure object, whose code is as specified by
the lambda and whose environment is E1, the environment in which
the lambda was evaluated to produce the procedure. e resulting pro-
cedure object is the value returned by the call to make-withdraw. is
is bound to W1 in the global environment, since the define itself is be-
ing evaluated in the global environment. Figure 3.7 shows the resulting
environment structure.
    Now we can analyze what happens when W1 is applied to an argu-
(W1 50)

   global        make-withdraw: ...
   env           W1:

                                                   Here is the balance
                           E1    balance: 100      that will be changed
                                                   by the set!

                                           amount: 50

   parameters: amount       (if (>= balance amount)
   body: ...                    (begin (set! balance
                                             (- balance amount))
                                "insufficient funds")

Figure 3.8: Environments created by applying the procedure object W1.

    We begin by constructing a frame in which amount, the formal pa-
rameter of W1, is bound to the argument 50. e crucial point to ob-
serve is that this frame has as its enclosing environment not the global
environment, but rather the environment E1, because this is the envi-
ronment that is specified by the W1 procedure object. Within this new
environment, we evaluate the body of the procedure:
(if (>= balance amount)
    (begin (set! balance (- balance amount))
    "Insufficient funds")

e resulting environment structure is shown in Figure 3.8. e expres-
sion being evaluated references both amount and balance. amount will
be found in the first frame in the environment, while balance will be

    global        make-withdraw: ...
    env           W1:

                            E1     balance: 50

             parameters: amount
             body: ...

             Figure 3.9: Environments aer the call to W1.

found by following the enclosing-environment pointer to E1.
    When the set! is executed, the binding of balance in E1 is changed.
At the completion of the call to W1, balance is 50, and the frame that
contains balance is still pointed to by the procedure object W1. e
frame that binds amount (in which we executed the code that changed
balance) is no longer relevant, since the procedure call that constructed
it has terminated, and there are no pointers to that frame from other
parts of the environment. e next time W1 is called, this will build a
new frame that binds amount and whose enclosing environment is E1.
We see that E1 serves as the “place” that holds the local state variable
for the procedure object W1. Figure 3.9 shows the situation aer the call
to W1.
    Observe what happens when we create a second “withdraw” object
by making another call to make-withdraw:
(define W2 (make-withdraw 100))

                      make-withdraw: ...

                       E1       balance: 50                 E2       balance: 100

          parameters: amount
          body: ...

       Figure 3.10: Using (define W2 (make-withdraw 100)) to
       create a second object.

    is produces the environment structure of Figure 3.10, which shows
that W2 is a procedure object, that is, a pair with some code and an en-
vironment. e environment E2 for W2 was created by the call to make-
withdraw. It contains a frame with its own local binding for balance.
On the other hand, W1 and W2 have the same code: the code specified
by the λ-expression in the body of make-withdraw.15 We see here why
W1 and W2 behave as independent objects. Calls to W1 reference the state
variable balance stored in E1, whereas calls to W2 reference the balance
stored in E2. us, changes to the local state of one object do not affect
the other object.

  15 Whether W1 and W2 share the same physical code stored in the computer, or whether

they each keep a copy of the code, is a detail of the implementation. For the interpreter
we implement in Chapter 4, the code is in fact shared.

Exercise 3.10: In the make-withdraw procedure, the local
variable balance is created as a parameter of make-withdraw.
We could also create the local state variable explicitly, us-
ing let, as follows:
(define (make-withdraw initial-amount)
  (let ((balance initial-amount))
    (lambda (amount)
      (if (>= balance amount)
           (begin (set! balance (- balance amount))
           "Insufficient funds"))))

Recall from Section 1.3.2 that let is simply syntactic sugar
for a procedure call:
(let ((⟨var⟩   ⟨exp⟩)) ⟨body⟩)

is interpreted as an alternate syntax for
((lambda (⟨var⟩)   ⟨body⟩) ⟨exp⟩)

Use the environment model to analyze this alternate ver-
sion of make-withdraw, drawing figures like the ones above
to illustrate the interactions
(define W1 (make-withdraw 100))
(W1 50)
(define W2 (make-withdraw 100))

Show that the two versions of make-withdraw create ob-
jects with the same behavior. How do the environment struc-
tures differ for the two versions?

3.2.4 Internal Definitions
Section 1.1.8 introduced the idea that procedures can have internal def-
initions, thus leading to a block structure as in the following procedure
to compute square roots:
(define (sqrt x)
  (define (good-enough? guess)
    (< (abs (- (square guess) x)) 0.001))
  (define (improve guess)
    (average guess (/ x guess)))
  (define (sqrt-iter guess)
    (if (good-enough? guess)
         (sqrt-iter (improve guess))))
  (sqrt-iter 1.0))

Now we can use the environment model to see why these internal defi-
nitions behave as desired. Figure 3.11 shows the point in the evaluation
of the expression (sqrt 2) where the internal procedure good-enough?
has been called for the first time with guess equal to 1.
    Observe the structure of the environment. sqrt is a symbol in the
global environment that is bound to a procedure object whose associ-
ated environment is the global environment. When sqrt was called, a
new environment E1 was formed, subordinate to the global environ-
ment, in which the parameter x is bound to 2. e body of sqrt was
then evaluated in E1. Since the first expression in the body of sqrt is
(define (good-enough? guess)
  (< (abs (- (square guess) x)) 0.001))

evaluating this expression defined the procedure good-enough? in the
environment E1. To be more precise, the symbol good-enough? was
added to the first frame of E1, bound to a procedure object whose asso-


                                             improve: ...
 parameters: x
                                             sqrt-iter: ...
 body: (define good-enough? ...)
          (define improve ...)
          (define sqrt-iter ...)
          (sqrt-iter 1.0)
                           E2      guess: 1
                                                      parameters: guess
                             call to sqrt-iter
                                                      body: (< (abs ...)

                                    E3        guess: 1

                                         call to good-enough?

      Figure 3.11: sqrt procedure with internal definitions.

ciated environment is E1. Similarly, improve and sqrt-iter were de-
fined as procedures in E1. For conciseness, Figure 3.11 shows only the
procedure object for good-enough?.
    Aer the local procedures were defined, the expression (sqrt-iter
1.0) was evaluated, still in environment E1. So the procedure object
bound to sqrt-iter in E1 was called with 1 as an argument. is cre-
ated an environment E2 in which guess, the parameter of sqrt-iter,
is bound to 1. sqrt-iter in turn called good-enough? with the value of
guess (from E2) as the argument for good-enough?. is set up another

environment, E3, in which guess (the parameter of good-enough?) is
bound to 1. Although sqrt-iter and good-enough? both have a pa-
rameter named guess, these are two distinct local variables located in
different frames. Also, E2 and E3 both have E1 as their enclosing en-
vironment, because the sqrt-iter and good-enough? procedures both
have E1 as their environment part. One consequence of this is that the
symbol x that appears in the body of good-enough? will reference the
binding of x that appears in E1, namely the value of x with which the
original sqrt procedure was called.
    e environment model thus explains the two key properties that
make local procedure definitions a useful technique for modularizing

    • e names of the local procedures do not interfere with names
      external to the enclosing procedure, because the local procedure
      names will be bound in the frame that the procedure creates when
      it is run, rather than being bound in the global environment.

    • e local procedures can access the arguments of the enclosing
      procedure, simply by using parameter names as free variables.
      is is because the body of the local procedure is evaluated in an
      environment that is subordinate to the evaluation environment
      for the enclosing procedure.

      Exercise 3.11: In Section 3.2.3 we saw how the environ-
      ment model described the behavior of procedures with local
      state. Now we have seen how internal definitions work. A
      typical message-passing procedure contains both of these
      aspects. Consider the bank account procedure of Section

(define (make-account balance)
     (define (withdraw amount)
      (if (>= balance amount)
          (begin (set! balance (- balance amount))
          "Insufficient funds"))
     (define (deposit amount)
      (set! balance (+ balance amount))
     (define (dispatch m)
      (cond ((eq? m 'withdraw) withdraw)
            ((eq? m 'deposit) deposit)
                 (error "Unknown request: MAKE-ACCOUNT"

Show the environment structure generated by the sequence
of interactions
(define acc (make-account 50))
((acc 'deposit) 40)
((acc 'withdraw) 60)

Where is the local state for acc kept? Suppose we define
another account
(define acc2 (make-account 100))

How are the local states for the two accounts kept distinct?
Which parts of the environment structure are shared be-
tween acc and acc2?

3.3 Modeling with Mutable Data
Chapter 2 dealt with compound data as a means for constructing com-
putational objects that have several parts, in order to model real-world
objects that have several aspects. In that chapter we introduced the dis-
cipline of data abstraction, according to which data structures are spec-
ified in terms of constructors, which create data objects, and selectors,
which access the parts of compound data objects. But we now know
that there is another aspect of data that Chapter 2 did not address. e
desire to model systems composed of objects that have changing state
leads us to the need to modify compound data objects, as well as to con-
struct and select from them. In order to model compound objects with
changing state, we will design data abstractions to include, in addition
to selectors and constructors, operations called mutators, which mod-
ify data objects. For instance, modeling a banking system requires us to
change account balances. us, a data structure for representing bank
accounts might admit an operation
(set-balance!   ⟨account⟩ ⟨new-value⟩)

that changes the balance of the designated account to the designated
new value. Data objects for which mutators are defined are known as
mutable data objects.
    Chapter 2 introduced pairs as a general-purpose “glue” for synthe-
sizing compound data. We begin this section by defining basic mutators
for pairs, so that pairs can serve as building blocks for constructing mu-
table data objects. ese mutators greatly enhance the representational
power of pairs, enabling us to build data structures other than the se-
quences and trees that we worked with in Section 2.2. We also present
some examples of simulations in which complex systems are modeled
as collections of objects with local state.

3.3.1 Mutable List Structure
e basic operations on pairs—cons, car, and cdr—can be used to con-
struct list structure and to select parts from list structure, but they are
incapable of modifying list structure. e same is true of the list oper-
ations we have used so far, such as append and list, since these can
be defined in terms of cons, car, and cdr. To modify list structures we
need new operations.
     e primitive mutators for pairs are set-car! and set-cdr!. set-
car! takes two arguments, the first of which must be a pair. It modifies
this pair, replacing the car pointer by a pointer to the second argument
of set-car!.16
     As an example, suppose that x is bound to the list ((a b) c d) and
y to the list (e f) as illustrated in Figure 3.12. Evaluating the expression
(set-car! x y) modifies the pair to which x is bound, replacing its car
by the value of y. e result of the operation is shown in Figure 3.13.
e structure x has been modified and would now be printed as ((e f)
c d). e pairs representing the list (a b), identified by the pointer that
was replaced, are now detached from the original structure.17
     Compare Figure 3.13 with Figure 3.14, which illustrates the result of
executing (define z (cons y (cdr x))) with x and y bound to the
original lists of Figure 3.12. e variable z is now bound to a new pair
created by the cons operation; the list to which x is bound is unchanged.
     e set-cdr! operation is similar to set-car!. e only difference
is that the cdr pointer of the pair, rather than the car pointer, is replaced.
e effect of executing (set-cdr! x y) on the lists of Figure 3.12 is
  16 set-car! and set-cdr! return implementation-dependent values. Like set!, they

should be used only for their effect.
  17 We see from this that mutation operations on lists can create “garbage” that is

not part of any accessible structure. We will see in Section 5.3.2 that Lisp memory-
management systems include a garbage collector, which identifies and recycles the mem-
ory space used by unneeded pairs.

                                    c           d

                                    a           b


                                    e           f

   Figure 3.12: Lists x: ((a b) c d) and y: (e f).


                                    c           d

                                    a           b


                                    e           f

Figure 3.13: Effect of (set-car! x y) on the lists in Figure 3.12.


                                    c           d

         z                          a           b


                                    e           f

   Figure 3.14: Effect of (define z (cons y (cdr x))) on
   the lists in Figure 3.12.


                                    c           d

                                    a           b


                                    e           f

Figure 3.15: Effect of (set-cdr! x y) on the lists in Figure 3.12.

shown in Figure 3.15. Here the cdr pointer of x has been replaced by
the pointer to (e f). Also, the list (c d), which used to be the cdr of
x, is now detached from the structure.
     cons builds new list structure by creating new pairs, while set-car!
and set-cdr! modify existing pairs. Indeed, we could implement cons
in terms of the two mutators, together with a procedure get-new-pair,
which returns a new pair that is not part of any existing list structure.
We obtain the new pair, set its car and cdr pointers to the designated
objects, and return the new pair as the result of the cons.18
(define (cons x y)
  (let ((new (get-new-pair)))
     (set-car! new x)
     (set-cdr! new y)

       Exercise 3.12: e following procedure for appending lists
       was introduced in Section 2.2.1:
       (define (append x y)
         (if (null? x)
              (cons (car x) (append (cdr x) y))))

       append   forms a new list by successively consing the el-
       ements of x onto y. e procedure append! is similar to
       append, but it is a mutator rather than a constructor. It ap-
       pends the lists by splicing them together, modifying the fi-
       nal pair of x so that its cdr is now y. (It is an error to call
       append! with an empty x.)
   18 get-new-pair is one of the operations that must be implemented as part of the

memory management required by a Lisp implementation. We will discuss this in Sec-
tion 5.3.1.

(define (append! x y)
    (set-cdr! (last-pair x) y)

Here last-pair is a procedure that returns the last pair in
its argument:
(define (last-pair x)
    (if (null? (cdr x)) x (last-pair (cdr x))))

Consider the interaction
(define x (list 'a 'b))
(define y (list 'c 'd))
(define z (append x y))
(a b c d)
(cdr x)
(define w (append! x y))
(a b c d)
(cdr x)

What are the missing ⟨response⟩s? Draw box-and-pointer
diagrams to explain your answer.

Exercise 3.13: Consider the following make-cycle proce-
dure, which uses the last-pair procedure defined in Exer-
cise 3.12:
(define (make-cycle x)
    (set-cdr! (last-pair x) x)

      Draw a box-and-pointer diagram that shows the structure
      z created by
      (define z (make-cycle (list 'a 'b 'c)))

      What happens if we try to compute (last-pair z)?

      Exercise 3.14: e following procedure is quite useful, al-
      though obscure:
      (define (mystery x)
        (define (loop x y)
          (if (null? x)
              (let ((temp (cdr x)))
                  (set-cdr! x y)
                  (loop temp x))))
        (loop x '()))

      loop  uses the “temporary” variable temp to hold the old
      value of the cdr of x, since the set-cdr! on the next line
      destroys the cdr. Explain what mystery does in general.
      Suppose v is defined by (define v (list 'a 'b 'c
      'd)). Draw the box-and-pointer diagram that represents
      the list to which v is bound. Suppose that we now evalu-
      ate (define w (mystery v)). Draw box-and-pointer dia-
      grams that show the structures v and w aer evaluating this
      expression. What would be printed as the values of v and

Sharing and identity
We mentioned in Section 3.1.3 the theoretical issues of “sameness” and
“change” raised by the introduction of assignment. ese issues arise in

practice when individual pairs are shared among different data objects.
For example, consider the structure formed by
(define x (list 'a 'b))
(define z1 (cons x x))

As shown in Figure 3.16, z1 is a pair whose car and cdr both point to
the same pair x. is sharing of x by the car and cdr of z1 is a con-
sequence of the straightforward way in which cons is implemented. In
general, using cons to construct lists will result in an interlinked struc-
ture of pairs in which many individual pairs are shared by many differ-
ent structures.
    In contrast to Figure 3.16, Figure 3.17 shows the structure created
(define z2 (cons (list 'a 'b) (list 'a 'b)))

In this structure, the pairs in the two (a b) lists are distinct, although
the actual symbols are shared.19
    When thought of as a list, z1 and z2 both represent “the same” list,
((a b) a b). In general, sharing is completely undetectable if we oper-
ate on lists using only cons, car, and cdr. However, if we allow mutators
on list structure, sharing becomes significant. As an example of the dif-
ference that sharing can make, consider the following procedure, which
modifies the car of the structure to which it is applied:
(define (set-to-wow! x) (set-car! (car x) 'wow) x)

   19 e two pairs are distinct because each call to cons returns a new pair. e symbols

are shared; in Scheme there is a unique symbol with any given name. Since Scheme
provides no way to mutate a symbol, this sharing is undetectable. Note also that the
sharing is what enables us to compare symbols using eq?, which simply checks equality
of pointers.



                               a             b

      Figure 3.16: e list z1 formed by (cons x x).


                                         a         b

      Figure 3.17: e list z2 formed by (cons (list 'a 'b)
      (list 'a 'b)).

Even though z1 and z2 are “the same” structure, applying set-to-wow!
to them yields different results. With z1, altering the car also changes
the cdr, because in z1 the car and the cdr are the same pair. With z2,
the car and cdr are distinct, so set-to-wow! modifies only the car:
((a b) a b)
(set-to-wow! z1)
((wow b) wow b)
((a b) a b)

(set-to-wow! z2)
((wow b) a b)

One way to detect sharing in list structures is to use the predicate eq?,
which we introduced in Section 2.3.1 as a way to test whether two sym-
bols are equal. More generally, (eq? x y) tests whether x and y are
the same object (that is, whether x and y are equal as pointers). us,
with z1 and z2 as defined in Figure 3.16 and Figure 3.17, (eq? (car z1)
(cdr z1)) is true and (eq? (car z2) (cdr z2)) is false.
    As will be seen in the following sections, we can exploit sharing to
greatly extend the repertoire of data structures that can be represented
by pairs. On the other hand, sharing can also be dangerous, since modi-
fications made to structures will also affect other structures that happen
to share the modified parts. e mutation operations set-car! and set-
cdr! should be used with care; unless we have a good understanding of
how our data objects are shared, mutation can have unanticipated re-
        Exercise 3.15: Draw box-and-pointer diagrams to explain
        the effect of set-to-wow! on the structures z1 and z2 above.
        Exercise 3.16: Ben Bitdiddle decides to write a procedure
        to count the number of pairs in any list structure. “It’s easy,”
   20 e subtleties of dealing with sharing of mutable data objects reflect the underlying

issues of “sameness” and “change” that were raised in Section 3.1.3. We mentioned there
that admiing change to our language requires that a compound object must have an
“identity” that is something different from the pieces from which it is composed. In
Lisp, we consider this “identity” to be the quality that is tested by eq?, i.e., by equality of
pointers. Since in most Lisp implementations a pointer is essentially a memory address,
we are “solving the problem” of defining the identity of objects by stipulating that a data
object “itself ” is the information stored in some particular set of memory locations in
the computer. is suffices for simple Lisp programs, but is hardly a general way to
resolve the issue of “sameness” in computational models.

he reasons. “e number of pairs in any structure is the
number in the car plus the number in the cdr plus one
more to count the current pair.” So Ben writes the following
(define (count-pairs x)
  (if (not (pair? x))
      (+ (count-pairs (car x))
          (count-pairs (cdr x))

Show that this procedure is not correct. In particular, draw
box-and-pointer diagrams representing list structures made
up of exactly three pairs for which Ben’s procedure would
return 3; return 4; return 7; never return at all.

Exercise 3.17: Devise a correct version of the count-pairs
procedure of Exercise 3.16 that returns the number of dis-
tinct pairs in any structure. (Hint: Traverse the structure,
maintaining an auxiliary data structure that is used to keep
track of which pairs have already been counted.)

Exercise 3.18: Write a procedure that examines a list and
determines whether it contains a cycle, that is, whether a
program that tried to find the end of the list by taking suc-
cessive cdrs would go into an infinite loop. Exercise 3.13
constructed such lists.

Exercise 3.19: Redo Exercise 3.18 using an algorithm that
takes only a constant amount of space. (is requires a very
clever idea.)

Mutation is just assignment
When we introduced compound data, we observed in Section 2.1.3 that
pairs can be represented purely in terms of procedures:
(define (cons x y)
  (define (dispatch m)
    (cond ((eq? m 'car) x)
          ((eq? m 'cdr) y)
          (else (error "Undefined operation: CONS" m))))
(define (car z) (z 'car))
(define (cdr z) (z 'cdr))

e same observation is true for mutable data. We can implement mu-
table data objects as procedures using assignment and local state. For
instance, we can extend the above pair implementation to handle set-
car! and set-cdr! in a manner analogous to the way we implemented
bank accounts using make-account in Section 3.1.1:
(define (cons x y)
  (define (set-x! v) (set! x v))
  (define (set-y! v) (set! y v))
  (define (dispatch m)
    (cond ((eq? m 'car) x)
          ((eq? m 'cdr) y)
          ((eq? m 'set-car!) set-x!)
          ((eq? m 'set-cdr!) set-y!)
           (error "Undefined operation: CONS" m))))
(define (car z) (z 'car))
(define (cdr z) (z 'cdr))
(define (set-car! z new-value)
  ((z 'set-car!) new-value) z)

(define (set-cdr! z new-value)
  ((z 'set-cdr!) new-value) z)

Assignment is all that is needed, theoretically, to account for the behav-
ior of mutable data. As soon as we admit set! to our language, we raise
all the issues, not only of assignment, but of mutable data in general.21

       Exercise 3.20: Draw environment diagrams to illustrate
       the evaluation of the sequence of expressions
       (define x (cons 1 2))
       (define z (cons x x))
       (set-car! (cdr z) 17)
       (car x)

       using the procedural implementation of pairs given above.
       (Compare Exercise 3.11.)

3.3.2 Representing eues
e mutators set-car! and set-cdr! enable us to use pairs to construct
data structures that cannot be built with cons, car, and cdr alone. is
section shows how to use pairs to represent a data structure called a
queue. Section 3.3.3 will show how to represent data structures called
    A queue is a sequence in which items are inserted at one end (called
the rear of the queue) and deleted from the other end (the front ). Fig-
ure 3.18 shows an initially empty queue in which the items a and b are
  21 On the other hand, from the viewpoint of implementation, assignment requires us

to modify the environment, which is itself a mutable data structure. us, assignment
and mutation are equipotent: Each can be implemented in terms of the other.

             Operation                     Resulting Queue

             (define q (make-queue))
             (insert-queue! q 'a)          a
             (insert-queue! q 'b)          a   b
             (delete-queue! q)             b
             (insert-queue! q 'c)          b   c
             (insert-queue! q 'd)          b   c d
             (delete-queue! q)             c   d

                   Figure 3.18: eue operations.

inserted. en a is removed, c and d are inserted, and b is removed. Be-
cause items are always removed in the order in which they are inserted,
a queue is sometimes called a FIFO (first in, first out) buffer.
    In terms of data abstraction, we can regard a queue as defined by
the following set of operations:

    • a constructor: (make-queue) returns an empty queue (a queue
      containing no items).

    • two selectors:
      (empty-queue?      ⟨queue⟩) tests if the queue is empty.
      (front-queue    ⟨queue⟩) returns the object at the front of the
      queue, signaling an error if the queue is empty; it does not modify
      the queue.

    • two mutators:
      (insert-queue!   ⟨queue⟩ ⟨item⟩) inserts the item at the rear of
      the queue and returns the modified queue as its value.

      (delete-queue!     ⟨queue⟩) removes the item at the front of the
      queue and returns the modified queue as its value, signaling an
      error if the queue is empty before the deletion.

Because a queue is a sequence of items, we could certainly represent it
as an ordinary list; the front of the queue would be the car of the list,
inserting an item in the queue would amount to appending a new ele-
ment at the end of the list, and deleting an item from the queue would
just be taking the cdr of the list. However, this representation is ineffi-
cient, because in order to insert an item we must scan the list until we
reach the end. Since the only method we have for scanning a list is by
successive cdr operations, this scanning requires Θ(n) steps for a list of
n items. A simple modification to the list representation overcomes this
disadvantage by allowing the queue operations to be implemented so
that they require Θ(1) steps; that is, so that the number of steps needed
is independent of the length of the queue.
     e difficulty with the list representation arises from the need to
scan to find the end of the list. e reason we need to scan is that, al-
though the standard way of representing a list as a chain of pairs read-
ily provides us with a pointer to the beginning of the list, it gives us
no easily accessible pointer to the end. e modification that avoids the
drawback is to represent the queue as a list, together with an additional
pointer that indicates the final pair in the list. at way, when we go to
insert an item, we can consult the rear pointer and so avoid scanning
the list.
     A queue is represented, then, as a pair of pointers, front-ptr and
rear-ptr, which indicate, respectively, the first and last pairs in an or-
dinary list. Since we would like the queue to be an identifiable object, we
can use cons to combine the two pointers. us, the queue itself will be
the cons of the two pointers. Figure 3.19 illustrates this representation.


                          front-ptr                rear-ptr

                      a               b        c

      Figure 3.19: Implementation of a queue as a list with front
      and rear pointers.

   To define the queue operations we use the following procedures,
which enable us to select and to modify the front and rear pointers of a
(define (front-ptr queue) (car queue))
(define (rear-ptr    queue) (cdr queue))
(define (set-front-ptr! queue item)
  (set-car! queue item))
(define (set-rear-ptr!     queue item)
  (set-cdr! queue item))

Now we can implement the actual queue operations. We will consider
a queue to be empty if its front pointer is the empty list:
(define (empty-queue? queue)
  (null? (front-ptr queue)))

e make-queue constructor returns, as an initially empty queue, a pair
whose car and cdr are both the empty list:
(define (make-queue) (cons '() '()))


                     front-ptr                              rear-ptr

                 a               b         c            d

      Figure 3.20: Result of using (insert-queue! q 'd) on the
      queue of Figure 3.19.

To select the item at the front of the queue, we return the car of the pair
indicated by the front pointer:
(define (front-queue queue)
  (if (empty-queue? queue)
      (error "FRONT called with an empty queue" queue)
      (car (front-ptr queue))))

To insert an item in a queue, we follow the method whose result is in-
dicated in Figure 3.20. We first create a new pair whose car is the item
to be inserted and whose cdr is the empty list. If the queue was initially
empty, we set the front and rear pointers of the queue to this new pair.
Otherwise, we modify the final pair in the queue to point to the new
pair, and also set the rear pointer to the new pair.
(define (insert-queue! queue item)
  (let ((new-pair (cons item '())))
    (cond ((empty-queue? queue)
            (set-front-ptr! queue new-pair)
            (set-rear-ptr! queue new-pair)



                    a               b             c              d

          Figure 3.21: Result of using (delete-queue! q) on the
          queue of Figure 3.20.

                (set-cdr! (rear-ptr queue) new-pair)
                (set-rear-ptr! queue new-pair)

To delete the item at the front of the queue, we merely modify the front
pointer so that it now points at the second item in the queue, which
can be found by following the cdr pointer of the first item (see Figure
(define (delete-queue! queue)
  (cond ((empty-queue? queue)
              (error "DELETE! called with an empty queue" queue))
              (else (set-front-ptr! queue (cdr (front-ptr queue)))

  22 Ifthe first item is the final item in the queue, the front pointer will be the empty
list aer the deletion, which will mark the queue as empty; we needn’t worry about
updating the rear pointer, which will still point to the deleted item, because empty-
queue? looks only at the front pointer.

Exercise 3.21: Ben Bitdiddle decides to test the queue im-
plementation described above. He types in the procedures
to the Lisp interpreter and proceeds to try them out:
(define q1 (make-queue))
(insert-queue! q1 'a)
((a) a)
(insert-queue! q1 'b)
((a b) b)
(delete-queue! q1)
((b) b)
(delete-queue! q1)
(() b)

“It’s all wrong!” he complains. “e interpreter’s response
shows that the last item is inserted into the queue twice.
And when I delete both items, the second b is still there,
so the queue isn’t empty, even though it’s supposed to be.”
Eva Lu Ator suggests that Ben has misunderstood what is
happening. “It’s not that the items are going into the queue
twice,” she explains. “It’s just that the standard Lisp printer
doesn’t know how to make sense of the queue representa-
tion. If you want to see the queue printed correctly, you’ll
have to define your own print procedure for queues.” Ex-
plain what Eva Lu is talking about. In particular, show why
Ben’s examples produce the printed results that they do.
Define a procedure print-queue that takes a queue as in-
put and prints the sequence of items in the queue.

Exercise 3.22: Instead of representing a queue as a pair of
pointers, we can build a queue as a procedure with local
state. e local state will consist of pointers to the begin-

       ning and the end of an ordinary list. us, the make-queue
       procedure will have the form
       (define (make-queue)
          (let ((front-ptr        ... )
                  (rear-ptr     . . . ))
             ⟨definitions of internal procedures⟩
             (define (dispatch m) . . .)

       Complete the definition of make-queue and provide imple-
       mentations of the queue operations using this representa-

       Exercise 3.23: A deque (“double-ended queue”) is a sequence
       in which items can be inserted and deleted at either the
       front or the rear. Operations on deques are the constructor
       make-deque, the predicate empty-deque?, selectors front-
       deque and rear-deque, mutators front-insert-deque!,
       rear-insert-deque!, front-delete-deque!, and rear-delete-
       deque!. Show how to represent deques using pairs, and
       give implementations of the operations.23 All operations
       should be accomplished in Θ(1) steps.

3.3.3 Representing Tables
When we studied various ways of representing sets in Chapter 2, we
mentioned in Section 2.3.3 the task of maintaining a table of records in-
dexed by identifying keys. In the implementation of data-directed pro-
gramming in Section 2.4.3, we made extensive use of two-dimensional
  23 Be careful not to make the interpreter try to print a structure that contains cycles.

(See Exercise 3.13.)



                          a     1         b    2        c     3

           Figure 3.22: A table represented as a headed list.

tables, in which information is stored and retrieved using two keys. Here
we see how to build tables as mutable list structures.
    We first consider a one-dimensional table, in which each value is
stored under a single key. We implement the table as a list of records,
each of which is implemented as a pair consisting of a key and the as-
sociated value. e records are glued together to form a list by pairs
whose cars point to successive records. ese gluing pairs are called
the backbone of the table. In order to have a place that we can change
when we add a new record to the table, we build the table as a headed list.
A headed list has a special backbone pair at the beginning, which holds
a dummy “record”—in this case the arbitrarily chosen symbol *table*.
Figure 3.22 shows the box-and-pointer diagram for the table
a:   1
b:   2
c:   3

To extract information from a table we use the lookup procedure, which
takes a key as argument and returns the associated value (or false if

there is no value stored under that key). lookup is defined in terms of the
assoc operation, which expects a key and a list of records as arguments.
Note that assoc never sees the dummy record. assoc returns the record
that has the given key as its car.24 lookup then checks to see that the
resulting record returned by assoc is not false, and returns the value
(the cdr) of the record.
(define (lookup key table)
  (let ((record (assoc key (cdr table))))
     (if record
           (cdr record)
(define (assoc key records)
  (cond ((null? records) false)
           ((equal? key (caar records)) (car records))
           (else (assoc key (cdr records)))))

To insert a value in a table under a specified key, we first use assoc
to see if there is already a record in the table with this key. If not, we
form a new record by consing the key with the value, and insert this at
the head of the table’s list of records, aer the dummy record. If there
already is a record with this key, we set the cdr of this record to the
designated new value. e header of the table provides us with a fixed
location to modify in order to insert the new record.25
(define (insert! key value table)
  (let ((record (assoc key (cdr table))))

  24 Because assoc    uses equal?, it can recognize keys that are symbols, numbers, or
list structure.
   25 us, the first backbone pair is the object that represents the table “itself”; that is,

a pointer to the table is a pointer to this pair. is same backbone pair always starts
the table. If we did not arrange things in this way, insert! would have to return a new
value for the start of the table when it added a new record.

    (if record
         (set-cdr! record value)
         (set-cdr! table
                    (cons (cons key value)
                           (cdr table)))))

To construct a new table, we simply create a list containing the symbol

(define (make-table)
  (list '*table*))

Two-dimensional tables
In a two-dimensional table, each value is indexed by two keys. We can
construct such a table as a one-dimensional table in which each key
identifies a subtable. Figure 3.23 shows the box-and-pointer diagram
for the table

math:    +:   43         letters:      a:   97
         -:   45                       b:   98
         *:   42

which has two subtables. (e subtables don’t need a special header
symbol, since the key that identifies the subtable serves this purpose.)
   When we look up an item, we use the first key to identify the correct
subtable. en we use the second key to identify the record within the
(define (lookup key-1 key-2 table)
  (let ((subtable
         (assoc key-1 (cdr table))))




                                           a   97    b   98


                            +         43   -   45    *   42

             Figure 3.23: A two-dimensional table.

(if subtable
    (let ((record
              (assoc key-2 (cdr subtable))))
         (if record
             (cdr record)

    To insert a new item under a pair of keys, we use assoc to see if
there is a subtable stored under the first key. If not, we build a new
subtable containing the single record (key-2, value) and insert it into
the table under the first key. If a subtable already exists for the first key,
we insert the new record into this subtable, using the insertion method
for one-dimensional tables described above:
(define (insert! key-1 key-2 value table)
  (let ((subtable (assoc key-1 (cdr table))))
    (if subtable
         (let ((record (assoc key-2 (cdr subtable))))
           (if record
                (set-cdr! record value)
                (set-cdr! subtable
                            (cons (cons key-2 value)
                                   (cdr subtable)))))
         (set-cdr! table
                     (cons (list key-1
                                   (cons key-2 value))
                            (cdr table)))))

Creating local tables
e lookup and insert! operations defined above take the table as an
argument. is enables us to use programs that access more than one ta-
ble. Another way to deal with multiple tables is to have separate lookup
and insert! procedures for each table. We can do this by representing
a table procedurally, as an object that maintains an internal table as part
of its local state. When sent an appropriate message, this “table object”
supplies the procedure with which to operate on the internal table. Here
is a generator for two-dimensional tables represented in this fashion:

(define (make-table)
 (let ((local-table (list '*table*)))
    (define (lookup key-1 key-2)
      (let ((subtable
             (assoc key-1 (cdr local-table))))
        (if subtable
             (let ((record
                     (assoc key-2 (cdr subtable))))
                 (if record (cdr record) false))
    (define (insert! key-1 key-2 value)
      (let ((subtable
             (assoc key-1 (cdr local-table))))
        (if subtable
             (let ((record
                     (assoc key-2 (cdr subtable))))
                 (if record
                    (set-cdr! record value)
                    (set-cdr! subtable
                              (cons (cons key-2 value)
                                       (cdr subtable)))))
             (set-cdr! local-table
                        (cons (list key-1 (cons key-2 value))
                              (cdr local-table)))))
    (define (dispatch m)
      (cond ((eq? m 'lookup-proc) lookup)
             ((eq? m 'insert-proc!) insert!)
             (else (error "Unknown operation: TABLE" m))))

Using make-table, we could implement the get and put operations
used in Section 2.4.3 for data-directed programming, as follows:

(define operation-table (make-table))
(define get (operation-table 'lookup-proc))
(define put (operation-table 'insert-proc!))

get takes as arguments two keys, and put takes as arguments two keys
and a value. Both operations access the same local table, which is en-
capsulated within the object created by the call to make-table.

      Exercise 3.24: In the table implementations above, the keys
      are tested for equality using equal? (called by assoc). is
      is not always the appropriate test. For instance, we might
      have a table with numeric keys in which we don’t need an
      exact match to the number we’re looking up, but only a
      number within some tolerance of it. Design a table con-
      structor make-table that takes as an argument a same-key?
      procedure that will be used to test “equality” of keys. make-
      table should return a dispatch procedure that can be used
      to access appropriate lookup and insert! procedures for a
      local table.

      Exercise 3.25: Generalizing one- and two-dimensional ta-
      bles, show how to implement a table in which values are
      stored under an arbitrary number of keys and different val-
      ues may be stored under different numbers of keys. e
      lookup and insert! procedures should take as input a list
      of keys used to access the table.

      Exercise 3.26: To search a table as implemented above, one
      needs to scan through the list of records. is is basically
      the unordered list representation of Section 2.3.3. For large
      tables, it may be more efficient to structure the table in a dif-
      ferent manner. Describe a table implementation where the

(key, value) records are organized using a binary tree, as-
suming that keys can be ordered in some way (e.g., numer-
ically or alphabetically). (Compare Exercise 2.66 of Chapter

Exercise 3.27: Memoization (also called tabulation) is a tech-
nique that enables a procedure to record, in a local table,
values that have previously been computed. is technique
can make a vast difference in the performance of a program.
A memoized procedure maintains a table in which values
of previous calls are stored using as keys the arguments
that produced the values. When the memoized procedure
is asked to compute a value, it first checks the table to see
if the value is already there and, if so, just returns that value.
Otherwise, it computes the new value in the ordinary way
and stores this in the table. As an example of memoization,
recall from Section 1.2.2 the exponential process for com-
puting Fibonacci numbers:
(define (fib n)
  (cond ((= n 0) 0)
         ((= n 1) 1)
         (else (+ (fib (- n 1)) (fib (- n 2))))))

e memoized version of the same procedure is
(define memo-fib
   (lambda (n)
      (cond ((= n 0) 0)
             ((= n 1) 1)
             (else (+ (memo-fib (- n 1))
                        (memo-fib (- n 2))))))))

      where the memoizer is defined as
      (define (memoize f)
        (let ((table (make-table)))
          (lambda (x)
             (let ((previously-computed-result
                    (lookup x table)))
               (or previously-computed-result
                   (let ((result (f x)))
                      (insert! x result table)

      Draw an environment diagram to analyze the computation
      of (memo-fib 3). Explain why memo-fib computes the n th
      Fibonacci number in a number of steps proportional to n.
      Would the scheme still work if we had simply defined memo-
      fib to be (memoize fib)?

3.3.4 A Simulator for Digital Circuits
Designing complex digital systems, such as computers, is an important
engineering activity. Digital systems are constructed by interconnect-
ing simple elements. Although the behavior of these individual elements
is simple, networks of them can have very complex behavior. Computer
simulation of proposed circuit designs is an important tool used by digi-
tal systems engineers. In this section we design a system for performing
digital logic simulations. is system typifies a kind of program called
an event-driven simulation, in which actions (“events”) trigger further
events that happen at a later time, which in turn trigger more events,
and so on.
     Our computational model of a circuit will be composed of objects
that correspond to the elementary components from which the circuit

                Inverter         And-gate            Or-gate

    Figure 3.24: Primitive functions in the digital logic simulator.

is constructed. ere are wires, which carry digital signals. A digital sig-
nal may at any moment have only one of two possible values, 0 and
1. ere are also various types of digital function boxes, which connect
wires carrying input signals to other output wires. Such boxes produce
output signals computed from their input signals. e output signal is
delayed by a time that depends on the type of the function box. For
example, an inverter is a primitive function box that inverts its input.
If the input signal to an inverter changes to 0, then one inverter-delay
later the inverter will change its output signal to 1. If the input signal to
an inverter changes to 1, then one inverter-delay later the inverter will
change its output signal to 0. We draw an inverter symbolically as in
Figure 3.24. An and-gate, also shown in Figure 3.24, is a primitive func-
tion box with two inputs and one output. It drives its output signal to
a value that is the logical and of the inputs. at is, if both of its input
signals become 1, then one and-gate-delay time later the and-gate will
force its output signal to be 1; otherwise the output will be 0. An or-gate
is a similar two-input primitive function box that drives its output sig-
nal to a value that is the logical or of the inputs. at is, the output will
become 1 if at least one of the input signals is 1; otherwise the output
will become 0.
     We can connect primitive functions together to construct more com-
plex functions. To accomplish this we wire the outputs of some function
boxes to the inputs of other function boxes. For example, the half-adder

            A                      D


                   Figure 3.25: A half-adder circuit.

circuit shown in Figure 3.25 consists of an or-gate, two and-gates, and
an inverter. It takes two input signals, A and B, and has two output sig-
nals, S and C. S will become 1 whenever precisely one of A and B is 1,
and C will become 1 whenever A and B are both 1. We can see from the
figure that, because of the delays involved, the outputs may be gener-
ated at different times. Many of the difficulties in the design of digital
circuits arise from this fact.
    We will now build a program for modeling the digital logic circuits
we wish to study. e program will construct computational objects
modeling the wires, which will “hold” the signals. Function boxes will
be modeled by procedures that enforce the correct relationships among
the signals.
    One basic element of our simulation will be a procedure make-wire,
which constructs wires. For example, we can construct six wires as fol-
(define a (make-wire))
(define b (make-wire))
(define c (make-wire))
(define d (make-wire))
(define e (make-wire))
(define s (make-wire))

We aach a function box to a set of wires by calling a procedure that
constructs that kind of box. e arguments to the constructor procedure
are the wires to be aached to the box. For example, given that we can
construct and-gates, or-gates, and inverters, we can wire together the
half-adder shown in Figure 3.25:
(or-gate a b d)
(and-gate a b c)
(inverter c e)
(and-gate d e s)

Beer yet, we can explicitly name this operation by defining a procedure
half-adder that constructs this circuit, given the four external wires to
be aached to the half-adder:
(define (half-adder a b s c)
  (let ((d (make-wire)) (e (make-wire)))
     (or-gate a b d)
     (and-gate a b c)
     (inverter c e)
     (and-gate d e s)

e advantage of making this definition is that we can use half-adder
itself as a building block in creating more complex circuits. Figure 3.26,
for example, shows a full-adder composed of two half-adders and an
or-gate.26 We can construct a full-adder as follows:
  26 A full-adder is a basic circuit element used in adding two binary numbers. Here

A and B are the bits at corresponding positions in the two numbers to be added, and

               A                                                    SUM
                            half-                        or         Cout

                       Figure 3.26: A full-adder circuit.

(define (full-adder a b c-in sum c-out)
  (let ((s (make-wire)) (c1 (make-wire)) (c2 (make-wire)))
     (half-adder b c-in s c1)
     (half-adder a s sum c2)
     (or-gate c1 c2 c-out)

Having defined full-adder as a procedure, we can now use it as a build-
ing block for creating still more complex circuits. (For example, see Ex-
ercise 3.30.)
    In essence, our simulator provides us with the tools to construct a
language of circuits. If we adopt the general perspective on languages
with which we approached the study of Lisp in Section 1.1, we can say
that the primitive function boxes form the primitive elements of the
language, that wiring boxes together provides a means of combination,
and that specifying wiring paerns as procedures serves as a means of

Cin is the carry bit from the addition one place to the right. e circuit generates SUM,
which is the sum bit in the corresponding position, and Cout , which is the carry bit to
be propagated to the le.

Primitive function boxes
e primitive function boxes implement the “forces” by which a change
in the signal on one wire influences the signals on other wires. To build
function boxes, we use the following operations on wires:

    • (get-signal ⟨ wire ⟩)
      returns the current value of the signal on the wire.

    • (set-signal! ⟨ wire ⟩ ⟨ new value ⟩)
      changes the value of the signal on the wire to the new value.

    • (add-action! ⟨ wire ⟩ ⟨ procedure of no arguments⟩)
      asserts that the designated procedure should be run whenever
      the signal on the wire changes value. Such procedures are the
      vehicles by which changes in the signal value on the wire are
      communicated to other wires.

In addition, we will make use of a procedure after-delay that takes a
time delay and a procedure to be run and executes the given procedure
aer the given delay.
    Using these procedures, we can define the primitive digital logic
functions. To connect an input to an output through an inverter, we use
add-action! to associate with the input wire a procedure that will be
run whenever the signal on the input wire changes value. e proce-
dure computes the logical-not of the input signal, and then, aer one
inverter-delay, sets the output signal to be this new value:

(define (inverter input output)
  (define (invert-input)
    (let ((new-value (logical-not (get-signal input))))

      (after-delay inverter-delay
                       (lambda () (set-signal! output new-value)))))
  (add-action! input invert-input) 'ok)
(define (logical-not s)
  (cond ((= s 0) 1)
         ((= s 1) 0)
         (else (error "Invalid signal" s))))

An and-gate is a lile more complex. e action procedure must be run
if either of the inputs to the gate changes. It computes the logical-
and (using a procedure analogous to logical-not) of the values of the
signals on the input wires and sets up a change to the new value to
occur on the output wire aer one and-gate-delay.
(define (and-gate a1 a2 output)
  (define (and-action-procedure)
    (let ((new-value
            (logical-and (get-signal a1) (get-signal a2))))
         (lambda () (set-signal! output new-value)))))
  (add-action! a1 and-action-procedure)
  (add-action! a2 and-action-procedure)

     Exercise 3.28: Define an or-gate as a primitive function
     box. Your or-gate constructor should be similar to and-

     Exercise 3.29: Another way to construct an or-gate is as
     a compound digital logic device, built from and-gates and
     inverters. Define a procedure or-gate that accomplishes

           A1 B1        C1   A2 B2        C2         A3 B3        C3     An Bn C = 0

             FA                FA                      FA                     FA

                   S1                S2                      S3        Cn-1        Sn

       Figure 3.27: A ripple-carry adder for n-bit numbers.

    this. What is the delay time of the or-gate in terms of and-
    gate-delay and inverter-delay?

    Exercise 3.30: Figure 3.27 shows a ripple-carry adder formed
    by stringing together n full-adders. is is the simplest form
    of parallel adder for adding two n-bit binary numbers. e
    inputs A 1 , A 2 , A 3 , . . ., An and B 1 , B 2 , B 3 , . . ., Bn are the
    two binary numbers to be added (each A k and B k is a 0 or
    a 1). e circuit generates S 1 , S 2 , S 3 , . . ., Sn , the n bits of
    the sum, and C, the carry from the addition. Write a proce-
    dure ripple-carry-adder that generates this circuit. e
    procedure should take as arguments three lists of n wires
    each—the A k , the B k , and the S k —and also another wire C.
    e major drawback of the ripple-carry adder is the need
    to wait for the carry signals to propagate. What is the delay
    needed to obtain the complete output from an n-bit ripple-
    carry adder, expressed in terms of the delays for and-gates,
    or-gates, and inverters?

Representing wires
A wire in our simulation will be a computational object with two local
state variables: a signal-value (initially taken to be 0) and a collec-
tion of action-procedures to be run when the signal changes value.
We implement the wire, using message-passing style, as a collection of
local procedures together with a dispatch procedure that selects the ap-
propriate local operation, just as we did with the simple bank-account
object in Section 3.1.1:
(define (make-wire)
  (let ((signal-value 0) (action-procedures '()))
    (define (set-my-signal! new-value)
      (if (not (= signal-value new-value))
           (begin (set! signal-value new-value)
                  (call-each action-procedures))
    (define (accept-action-procedure! proc)
      (set! action-procedures
             (cons proc action-procedures))
    (define (dispatch m)
      (cond ((eq? m 'get-signal) signal-value)
             ((eq? m 'set-signal!) set-my-signal!)
             ((eq? m 'add-action!) accept-action-procedure!)
             (else (error "Unknown operation: WIRE" m))))

e local procedure set-my-signal! tests whether the new signal value
changes the signal on the wire. If so, it runs each of the action proce-
dures, using the following procedure call-each, which calls each of the
items in a list of no-argument procedures:
(define (call-each procedures)

  (if (null? procedures)
        (begin ((car procedures))
                 (call-each (cdr procedures)))))

e local procedure accept-action-procedure! adds the given proce-
dure to the list of procedures to be run, and then runs the new procedure
once. (See Exercise 3.31.)
    With the local dispatch procedure set up as specified, we can pro-
vide the following procedures to access the local operations on wires:27
(define (get-signal wire) (wire 'get-signal))
(define (set-signal! wire new-value)
  ((wire 'set-signal!) new-value))
(define (add-action! wire action-procedure)
  ((wire 'add-action!) action-procedure))

Wires, which have time-varying signals and may be incrementally at-
tached to devices, are typical of mutable objects. We have modeled them
as procedures with local state variables that are modified by assignment.
When a new wire is created, a new set of state variables is allocated
(by the let expression in make-wire) and a new dispatch procedure
is constructed and returned, capturing the environment with the new
state variables.
  27  ese procedures are simply syntactic sugar that allow us to use ordinary pro-
cedural syntax to access the local procedures of objects. It is striking that we can in-
terchange the role of “procedures” and “data” in such a simple way. For example, if we
write (wire 'get-signal) we think of wire as a procedure that is called with the mes-
sage get-signal as input. Alternatively, writing (get-signal wire) encourages us to
think of wire as a data object that is the input to a procedure get-signal. e truth of
the maer is that, in a language in which we can deal with procedures as objects, there
is no fundamental difference between “procedures” and “data,” and we can choose our
syntactic sugar to allow us to program in whatever style we choose.

    e wires are shared among the various devices that have been con-
nected to them. us, a change made by an interaction with one device
will affect all the other devices aached to the wire. e wire communi-
cates the change to its neighbors by calling the action procedures pro-
vided to it when the connections were established.

The agenda
e only thing needed to complete the simulator is after-delay. e
idea here is that we maintain a data structure, called an agenda, that
contains a schedule of things to do. e following operations are defined
for agendas:

    • (make-agenda) returns a new empty agenda.

    • (empty-agenda?      ⟨ agenda ⟩) is true if the specified agenda is

    • (first-agenda-item      ⟨ agenda ⟩) returns the first item on the

    • (remove-first-agenda-item! ⟨ agenda ⟩) modifies the agenda
      by removing the first item.

    • (add-to-agenda! ⟨ time ⟩ ⟨ action ⟩ ⟨ agenda ⟩) modifies the
      agenda by adding the given action procedure to be run at the spec-
      ified time.

    • (current-time ⟨ agenda ⟩) returns the current simulation time.

e particular agenda that we use is denoted by the-agenda. e pro-
cedure after-delay adds new elements to the-agenda:

(define (after-delay delay action)
  (add-to-agenda! (+ delay (current-time the-agenda))

e simulation is driven by the procedure propagate, which operates
on the-agenda, executing each procedure on the agenda in sequence. In
general, as the simulation runs, new items will be added to the agenda,
and propagate will continue the simulation as long as there are items
on the agenda:
(define (propagate)
  (if (empty-agenda? the-agenda)
      (let ((first-item (first-agenda-item the-agenda)))
        (remove-first-agenda-item! the-agenda)

A sample simulation
e following procedure, which places a “probe” on a wire, shows the
simulator in action. e probe tells the wire that, whenever its signal
changes value, it should print the new signal value, together with the
current time and a name that identifies the wire:
(define (probe name wire)
  (add-action! wire
                (lambda ()
                  (display name) (display " ")
                  (display (current-time the-agenda))
                  (display "   New-value = ")
                  (display (get-signal wire)))))

We begin by initializing the agenda and specifying delays for the prim-
itive function boxes:
(define the-agenda (make-agenda))
(define inverter-delay 2)
(define and-gate-delay 3)
(define or-gate-delay 5)

Now we define four wires, placing probes on two of them:
(define input-1 (make-wire))
(define input-2 (make-wire))
(define sum (make-wire))
(define carry (make-wire))

(probe 'sum sum)
sum 0 New-value = 0

(probe 'carry carry)
carry 0 New-value = 0

Next we connect the wires in a half-adder circuit (as in Figure 3.25), set
the signal on input-1 to 1, and run the simulation:
(half-adder input-1 input-2 sum carry)

(set-signal! input-1 1)

sum 8 New-value = 1

e sum signal changes to 1 at time 8. We are now eight time units from
the beginning of the simulation. At this point, we can set the signal on
input-2 to 1 and allow the values to propagate:

(set-signal! input-2 1)

carry 11 New-value = 1
sum 16 New-value = 0

e carry changes to 1 at time 11 and the sum changes to 0 at time 16.

       Exercise 3.31: e internal procedure accept-action-procedure!
       defined in make-wire specifies that when a new action pro-
       cedure is added to a wire, the procedure is immediately
       run. Explain why this initialization is necessary. In particu-
       lar, trace through the half-adder example in the paragraphs
       above and say how the system’s response would differ if we
       had defined accept-action-procedure! as
       (define (accept-action-procedure! proc)
         (set! action-procedures
               (cons proc action-procedures)))

Implementing the agenda
Finally, we give details of the agenda data structure, which holds the
procedures that are scheduled for future execution.
    e agenda is made up of time segments. Each time segment is a
pair consisting of a number (the time) and a queue (see Exercise 3.32)
that holds the procedures that are scheduled to be run during that time
(define (make-time-segment time queue)
  (cons time queue))

(define (segment-time s) (car s))
(define (segment-queue s) (cdr s))

We will operate on the time-segment queues using the queue operations
described in Section 3.3.2.
    e agenda itself is a one-dimensional table of time segments. It
differs from the tables described in Section 3.3.3 in that the segments will
be sorted in order of increasing time. In addition, we store the current
time (i.e., the time of the last action that was processed) at the head of
the agenda. A newly constructed agenda has no time segments and has
a current time of 0:28
(define (make-agenda) (list 0))
(define (current-time agenda) (car agenda))
(define (set-current-time! agenda time)
  (set-car! agenda time))
(define (segments agenda) (cdr agenda))
(define (set-segments! agenda segments)
  (set-cdr! agenda segments))
(define (first-segment agenda) (car (segments agenda)))
(define (rest-segments agenda) (cdr (segments agenda)))

An agenda is empty if it has no time segments:
(define (empty-agenda? agenda)
  (null? (segments agenda)))

To add an action to an agenda, we first check if the agenda is empty. If so,
we create a time segment for the action and install this in the agenda.
Otherwise, we scan the agenda, examining the time of each segment.
If we find a segment for our appointed time, we add the action to the
  28 e agenda is a headed list, like the tables in Section 3.3.3, but since the list is

headed by the time, we do not need an additional dummy header (such as the *table*
symbol used with tables).

associated queue. If we reach a time later than the one to which we are
appointed, we insert a new time segment into the agenda just before it.
If we reach the end of the agenda, we must create a new time segment
at the end.
(define (add-to-agenda! time action agenda)
  (define (belongs-before? segments)
    (or (null? segments)
         (< time (segment-time (car segments)))))
  (define (make-new-time-segment time action)
    (let ((q (make-queue)))
      (insert-queue! q action)
      (make-time-segment time q)))
  (define (add-to-segments! segments)
    (if (= (segment-time (car segments)) time)
         (insert-queue! (segment-queue (car segments))
         (let ((rest (cdr segments)))
           (if (belongs-before? rest)
                   (cons (make-new-time-segment time action)
                        (cdr segments)))
               (add-to-segments! rest)))))
  (let ((segments (segments agenda)))
    (if (belongs-before? segments)
          (cons (make-new-time-segment time action)
         (add-to-segments! segments))))

e procedure that removes the first item from the agenda deletes the
item at the front of the queue in the first time segment. If this deletion

makes the time segment empty, we remove it from the list of segments:29
(define (remove-first-agenda-item! agenda)
  (let ((q (segment-queue (first-segment agenda))))
     (delete-queue! q)
     (if (empty-queue? q)
           (set-segments! agenda (rest-segments agenda)))))

e first agenda item is found at the head of the queue in the first
time segment. Whenever we extract an item, we also update the cur-
rent time:30
(define (first-agenda-item agenda)
  (if (empty-agenda? agenda)
        (error "Agenda is empty: FIRST-AGENDA-ITEM")
        (let ((first-seg (first-segment agenda)))
           (set-current-time! agenda
                                     (segment-time first-seg))
           (front-queue (segment-queue first-seg)))))

       Exercise 3.32: e procedures to be run during each time
       segment of the agenda are kept in a queue. us, the pro-
       cedures for each segment are called in the order in which
       they were added to the agenda (first in, first out). Explain
       why this order must be used. In particular, trace the behav-
       ior of an and-gate whose inputs change from 0, 1 to 1, 0
  29 Observe    that the if expression in this procedure has no ⟨alternative ⟩ expression.
Such a “one-armed if statement” is used to decide whether to do something, rather
than to select between two expressions. An if expression returns an unspecified value
if the predicate is false and there is no ⟨alternative ⟩.
    30 In this way, the current time will always be the time of the action most recently

processed. Storing this time at the head of the agenda ensures that it will still be avail-
able even if the associated time segment has been deleted.

       in the same segment and say how the behavior would dif-
       fer if we stored a segment’s procedures in an ordinary list,
       adding and removing procedures only at the front (last in,
       first out).

3.3.5 Propagation of Constraints
Computer programs are traditionally organized as one-directional com-
putations, which perform operations on prespecified arguments to pro-
duce desired outputs. On the other hand, we oen model systems in
terms of relations among quantities. For example, a mathematical model
of a mechanical structure might include the information that the deflec-
tion d of a metal rod is related to the force F on the rod, the length L
of the rod, the cross-sectional area A, and the elastic modulus E via the
                                dAE = F L.
Such an equation is not one-directional. Given any four of the quanti-
ties, we can use it to compute the fih. Yet translating the equation into
a traditional computer language would force us to choose one of the
quantities to be computed in terms of the other four. us, a procedure
for computing the area A could not be used to compute the deflection
d, even though the computations of A and d arise from the same equa-
   31 Constraint propagation first appeared in the incredibly forward-looking 

 system of Ivan Sutherland (1963). A beautiful constraint-propagation system based
on the Smalltalk language was developed by Alan Borning (1977) at Xerox Palo Alto
Research Center. Sussman, Stallman, and Steele applied constraint propagation to elec-
trical circuit analysis (Sussman and Stallman 1975; Sussman and Steele 1980). TK!Solver
(Konopasek and Jayaraman 1984) is an extensive modeling environment based on

     In this section, we sketch the design of a language that enables us
to work in terms of relations themselves. e primitive elements of the
language are primitive constraints, which state that certain relations hold
between quantities. For example, (adder a b c) specifies that the quan-
tities a, b, and c must be related by the equation a + b = c, (multiplier
x y z) expresses the constraint xy = z, and (constant 3.14 x) says
that the value of x must be 3.14.
     Our language provides a means of combining primitive constraints
in order to express more complex relations. We combine constraints
by constructing constraint networks, in which constraints are joined by
connectors. A connector is an object that “holds” a value that may par-
ticipate in one or more constraints. For example, we know that the re-
lationship between Fahrenheit and Celsius temperatures is

                             9C = 5(F − 32).

Such a constraint can be thought of as a network consisting of primitive
adder, multiplier, and constant constraints (Figure 3.28). In the figure,
we see on the le a multiplier box with three terminals, labeled m1,
m2, and p. ese connect the multiplier to the rest of the network as
follows: e m1 terminal is linked to a connector C, which will hold the
Celsius temperature. e m2 terminal is linked to a connector w, which
is also linked to a constant box that holds 9. e p terminal, which the
multiplier box constrains to be the product of m1 and m2, is linked to
the p terminal of another multiplier box, whose m2 is connected to a
constant 5 and whose m1 is connected to one of the terms in a sum.
     Computation by such a network proceeds as follows: When a con-
nector is given a value (by the user or by a constraint box to which
it is linked), it awakens all of its associated constraints (except for the
constraint that just awakened it) to inform them that it has a value.

       C     m1                          m1               a1
                  *   p        p   *                           +    s   F
             m2                          m2               a2

       w                                      x       y
                  9                 5                          32

      Figure 3.28: e relation 9C = 5(F − 32) expressed as a
      constraint network.

Each awakened constraint box then polls its connectors to see if there
is enough information to determine a value for a connector. If so, the
box sets that connector, which then awakens all of its associated con-
straints, and so on. For instance, in conversion between Celsius and
Fahrenheit, w, x , and y are immediately set by the constant boxes to 9,
5, and 32, respectively. e connectors awaken the multipliers and the
adder, which determine that there is not enough information to pro-
ceed. If the user (or some other part of the network) sets C to a value
(say 25), the lemost multiplier will be awakened, and it will set u to
25 · 9 = 225. en u awakens the second multiplier, which sets v to 45,
and v awakens the adder, which sets f to 77.

Using the constraint system
To use the constraint system to carry out the temperature computation
outlined above, we first create two connectors, C and F, by calling the
constructor make-connector, and link C and F in an appropriate net-
(define C (make-connector))
(define F (make-connector))

(celsius-fahrenheit-converter C F)

e procedure that creates the network is defined as follows:
(define (celsius-fahrenheit-converter c f)
  (let ((u (make-connector))
         (v (make-connector))
         (w (make-connector))
         (x (make-connector))
         (y (make-connector)))
     (multiplier c w u)
     (multiplier v x u)
     (adder v y f)
     (constant 9 w)
     (constant 5 x)
     (constant 32 y)

is procedure creates the internal connectors u, v, w, x, and y, and links
them as shown in Figure 3.28 using the primitive constraint construc-
tors adder, multiplier, and constant. Just as with the digital-circuit
simulator of Section 3.3.4, expressing these combinations of primitive
elements in terms of procedures automatically provides our language
with a means of abstraction for compound objects.
    To watch the network in action, we can place probes on the con-
nectors C and F, using a probe procedure similar to the one we used to
monitor wires in Section 3.3.4. Placing a probe on a connector will cause
a message to be printed whenever the connector is given a value:
(probe "Celsius temp" C)
(probe "Fahrenheit temp" F)

Next we set the value of C to 25. (e third argument to set-value!
tells C that this directive comes from the user.)

(set-value! C 25 'user)
Probe: Celsius temp = 25
Probe: Fahrenheit temp = 77

e probe on C awakens and reports the value. C also propagates its
value through the network as described above. is sets F to 77, which
is reported by the probe on F.
     Now we can try to set F to a new value, say 212:
(set-value! F 212 'user)
Error! Contradiction (77 212)

e connector complains that it has sensed a contradiction: Its value is
77, and someone is trying to set it to 212. If we really want to reuse the
network with new values, we can tell C to forget its old value:
(forget-value! C 'user)
Probe: Celsius temp = ?
Probe: Fahrenheit temp = ?

C finds that the user, who set its value originally, is now retracting that
value, so C agrees to lose its value, as shown by the probe, and informs
the rest of the network of this fact. is information eventually prop-
agates to F, which now finds that it has no reason for continuing to
believe that its own value is 77. us, F also gives up its value, as shown
by the probe.
    Now that F has no value, we are free to set it to 212:
(set-value! F 212 'user)
Probe: Fahrenheit temp = 212
Probe: Celsius temp = 100

is new value, when propagated through the network, forces C to have
a value of 100, and this is registered by the probe on C. Notice that the
very same network is being used to compute C given F and to compute
F given C. is nondirectionality of computation is the distinguishing
feature of constraint-based systems.

Implementing the constraint system
e constraint system is implemented via procedural objects with local
state, in a manner very similar to the digital-circuit simulator of Sec-
tion 3.3.4. Although the primitive objects of the constraint system are
somewhat more complex, the overall system is simpler, since there is
no concern about agendas and logic delays.
    e basic operations on connectors are the following:

    • (has-value? ⟨connector⟩) tells whether the connector has a

    • (get-value ⟨connector⟩) returns the connector’s current value.

    • (set-value! ⟨connector⟩ ⟨new-value⟩ ⟨informant⟩) indicates
      that the informant is requesting the connector to set its value to
      the new value.

    • (forget-value! ⟨connector⟩ ⟨retractor⟩) tells the connector
      that the retractor is requesting it to forget its value.

    • (connect ⟨connector⟩ ⟨new-constraint⟩) tells the connector
      to participate in the new constraint.

e connectors communicate with the constraints by means of the pro-
cedures inform-about-value, which tells the given constraint that the

connector has a value, and inform-about-no-value, which tells the
constraint that the connector has lost its value.
    adder constructs an adder constraint among summand connectors
a1 and a2 and a sum connector. An adder is implemented as a procedure
with local state (the procedure me below):
(define (adder a1 a2 sum)
  (define (process-new-value)
    (cond ((and (has-value? a1) (has-value? a2))
           (set-value! sum
                         (+ (get-value a1) (get-value a2))
          ((and (has-value? a1) (has-value? sum))
           (set-value! a2
                         (- (get-value sum) (get-value a1))
          ((and (has-value? a2) (has-value? sum))
           (set-value! a1
                         (- (get-value sum) (get-value a2))
  (define (process-forget-value)
    (forget-value! sum me)
    (forget-value! a1 me)
    (forget-value! a2 me)
  (define (me request)
    (cond ((eq? request 'I-have-a-value)    (process-new-value))
          ((eq? request 'I-lost-my-value) (process-forget-value))
          (else (error "Unknown request: ADDER" request))))
  (connect a1 me)
  (connect a2 me)
  (connect sum me)

adder connects the new adder to the designated connectors and returns
it as its value. e procedure me, which represents the adder, acts as a
dispatch to the local procedures. e following “syntax interfaces” (see
Footnote 27 in Section 3.3.4) are used in conjunction with the dispatch:
(define (inform-about-value constraint)
  (constraint 'I-have-a-value))
(define (inform-about-no-value constraint)
  (constraint 'I-lost-my-value))

e adder’s local procedure process-new-value is called when the adder
is informed that one of its connectors has a value. e adder first checks
to see if both a1 and a2 have values. If so, it tells sum to set its value to
the sum of the two addends. e informant argument to set-value! is
me, which is the adder object itself. If a1 and a2 do not both have values,
then the adder checks to see if perhaps a1 and sum have values. If so, it
sets a2 to the difference of these two. Finally, if a2 and sum have values,
this gives the adder enough information to set a1. If the adder is told
that one of its connectors has lost a value, it requests that all of its con-
nectors now lose their values. (Only those values that were set by this
adder are actually lost.) en it runs process-new-value. e reason
for this last step is that one or more connectors may still have a value
(that is, a connector may have had a value that was not originally set by
the adder), and these values may need to be propagated back through
the adder.
     A multiplier is very similar to an adder. It will set its product to 0 if
either of the factors is 0, even if the other factor is not known.
(define (multiplier m1 m2 product)
  (define (process-new-value)
    (cond ((or (and (has-value? m1) (= (get-value m1) 0))
                 (and (has-value? m2) (= (get-value m2) 0)))

           (set-value! product 0 me))
          ((and (has-value? m1) (has-value? m2))
           (set-value! product
                         (* (get-value m1) (get-value m2))
          ((and (has-value? product) (has-value? m1))
           (set-value! m2
                         (/ (get-value product)
                            (get-value m1))
          ((and (has-value? product) (has-value? m2))
           (set-value! m1
                         (/ (get-value product)
                            (get-value m2))
  (define (process-forget-value)
    (forget-value! product me)
    (forget-value! m1 me)
    (forget-value! m2 me)
  (define (me request)
    (cond ((eq? request 'I-have-a-value)      (process-new-value))
          ((eq? request 'I-lost-my-value) (process-forget-value))
          (else (error "Unknown request: MULTIPLIER"
  (connect m1 me)
  (connect m2 me)
  (connect product me)

A constant constructor simply sets the value of the designated con-
nector. Any I-have-a-value or I-lost-my-value message sent to the
constant box will produce an error.

(define (constant value connector)
  (define (me request)
    (error "Unknown request: CONSTANT" request))
  (connect connector me)
  (set-value! connector value me)

Finally, a probe prints a message about the seing or unseing of the
designated connector:
(define (probe name connector)
  (define (print-probe value)
    (newline) (display "Probe: ") (display name)
    (display " = ") (display value))
  (define (process-new-value)
    (print-probe (get-value connector)))
  (define (process-forget-value) (print-probe "?"))
  (define (me request)
    (cond ((eq? request 'I-have-a-value)        (process-new-value))
           ((eq? request 'I-lost-my-value) (process-forget-value))
           (else (error "Unknown request: PROBE" request))))
  (connect connector me)

Representing connectors
A connector is represented as a procedural object with local state vari-
ables value, the current value of the connector; informant, the object
that set the connector’s value; and constraints, a list of the constraints
in which the connector participates.
(define (make-connector)
  (let ((value false) (informant false) (constraints '()))
    (define (set-my-value newval setter)

 (cond ((not (has-value? me))
          (set! value newval)
          (set! informant setter)
          (for-each-except setter
        ((not (= value newval))
          (error "Contradiction" (list value newval)))
        (else 'ignored)))
(define (forget-my-value retractor)
 (if (eq? retractor informant)
       (begin (set! informant false)
              (for-each-except retractor
(define (connect new-constraint)
 (if (not (memq new-constraint constraints))
       (set! constraints
             (cons new-constraint constraints)))
 (if (has-value? me)
       (inform-about-value new-constraint))
(define (me request)
 (cond ((eq? request 'has-value?)
          (if informant true false))
        ((eq? request 'value) value)
        ((eq? request 'set-value!) set-my-value)
        ((eq? request 'forget) forget-my-value)
        ((eq? request 'connect) connect)
        (else (error "Unknown operation: CONNECTOR"

e connector’s local procedure set-my-value is called when there is
a request to set the connector’s value. If the connector does not cur-
rently have a value, it will set its value and remember as informant
the constraint that requested the value to be set.32 en the connector
will notify all of its participating constraints except the constraint that
requested the value to be set. is is accomplished using the follow-
ing iterator, which applies a designated procedure to all items in a list
except a given one:
(define (for-each-except exception procedure list)
  (define (loop items)
     (cond ((null? items) 'done)
             ((eq? (car items) exception) (loop (cdr items)))
             (else (procedure (car items))
                     (loop (cdr items)))))
  (loop list))

If a connector is asked to forget its value, it runs the local procedure
forget-my-value, which first checks to make sure that the request is
coming from the same object that set the value originally. If so, the con-
nector informs its associated constraints about the loss of the value.
    e local procedure connect adds the designated new constraint to
the list of constraints if it is not already in that list. en, if the connector
has a value, it informs the new constraint of this fact.
    e connector’s procedure me serves as a dispatch to the other in-
ternal procedures and also represents the connector as an object. e
following procedures provide a syntax interface for the dispatch:
(define (has-value? connector)
  (connector 'has-value?))

  32 e setter   might not be a constraint. In our temperature example, we used user
as the setter.

(define (get-value connector)
 (connector 'value))
(define (set-value! connector new-value informant)
 ((connector 'set-value!) new-value informant))
(define (forget-value! connector retractor)
 ((connector 'forget) retractor))
(define (connect connector new-constraint)
 ((connector 'connect) new-constraint))

     Exercise 3.33: Using primitive multiplier, adder, and con-
     stant constraints, define a procedure averager that takes
     three connectors a, b, and c as inputs and establishes the
     constraint that the value of c is the average of the values of
     a and b.

     Exercise 3.34: Louis Reasoner wants to build a squarer, a
     constraint device with two terminals such that the value
     of connector b on the second terminal will always be the
     square of the value a on the first terminal. He proposes the
     following simple device made from a multiplier:
     (define (squarer a b)
       (multiplier a a b))

     ere is a serious flaw in this idea. Explain.

     Exercise 3.35: Ben Bitdiddle tells Louis that one way to
     avoid the trouble in Exercise 3.34 is to define a squarer
     as a new primitive constraint. Fill in the missing portions
     in Ben’s outline for a procedure to implement such a con-

(define (squarer a b)
  (define (process-new-value)
    (if (has-value? b)
        (if (< (get-value b) 0)
             (error "square less than 0: SQUARER"
                    (get-value b))
  (define (process-forget-value)    ⟨body1⟩)
  (define (me request)   ⟨body2⟩)
  ⟨rest of definition⟩

Exercise 3.36: Suppose we evaluate the following sequence
of expressions in the global environment:
(define a (make-connector))
(define b (make-connector))
(set-value! a 10 'user)

At some time during evaluation of the set-value!, the fol-
lowing expression from the connector’s local procedure is
  setter inform-about-value constraints)

Draw an environment diagram showing the environment
in which the above expression is evaluated.

Exercise 3.37: e celsius-fahrenheit-converter pro-
cedure is cumbersome when compared with a more expression-
oriented style of definition, such as

       (define (celsius-fahrenheit-converter x)
          (c+ (c* (c/ (cv 9) (cv 5))
               (cv 32)))
       (define C (make-connector))
       (define F (celsius-fahrenheit-converter C))

       Here c+, c*, etc. are the “constraint” versions of the arith-
       metic operations. For example, c+ takes two connectors as
       arguments and returns a connector that is related to these
       by an adder constraint:
       (define (c+ x y)
          (let ((z (make-connector)))
             (adder x y z)

       Define analogous procedures c-, c*, c/, and cv (constant
       value) that enable us to define compound constraints as in
       the converter example above.33
   33 e expression-oriented format is convenient because it avoids the need to name

the intermediate expressions in a computation. Our original formulation of the con-
straint language is cumbersome in the same way that many languages are cumbersome
when dealing with operations on compound data. For example, if we wanted to com-
pute the product (a +b) · (c +d), where the variables represent vectors, we could work in
“imperative style,” using procedures that set the values of designated vector arguments
but do not themselves return vectors as values:
(v-sum a b temp1)
(v-sum c d temp2)
(v-prod temp1 temp2 answer)

Alternatively, we could deal with expressions, using procedures that return vectors as
values, and thus avoid explicitly mentioning temp1 and temp2:
(define answer (v-prod (v-sum a b) (v-sum c d)))

3.4 Concurrency: Time Is of the Essence
We’ve seen the power of computational objects with local state as tools
for modeling. Yet, as Section 3.1.3 warned, this power extracts a price:
the loss of referential transparency, giving rise to a thicket of questions
about sameness and change, and the need to abandon the substitution
model of evaluation in favor of the more intricate environment model.
    e central issue lurking beneath the complexity of state, sameness,
and change is that by introducing assignment we are forced to admit
time into our computational models. Before we introduced assignment,
all our programs were timeless, in the sense that any expression that
has a value always has the same value. In contrast, recall the example of
modeling withdrawals from a bank account and returning the resulting
balance, introduced at the beginning of Section 3.1.1:
(withdraw 25)
(withdraw 25)

Since Lisp allows us to return compound objects as values of procedures, we can trans-
form our imperative-style constraint language into an expression-oriented style as
shown in this exercise. In languages that are impoverished in handling compound ob-
jects, such as Algol, Basic, and Pascal (unless one explicitly uses Pascal pointer vari-
ables), one is usually stuck with the imperative style when manipulating compound
objects. Given the advantage of the expression-oriented format, one might ask if there
is any reason to have implemented the system in imperative style, as we did in this
section. One reason is that the non-expression-oriented constraint language provides
a handle on constraint objects (e.g., the value of the adder procedure) as well as on
connector objects. is is useful if we wish to extend the system with new operations
that communicate with constraints directly rather than only indirectly via operations
on connectors. Although it is easy to implement the expression-oriented style in terms
of the imperative implementation, it is very difficult to do the converse.

Here successive evaluations of the same expression yield different val-
ues. is behavior arises from the fact that the execution of assignment
statements (in this case, assignments to the variable balance) delineates
moments in time when values change. e result of evaluating an ex-
pression depends not only on the expression itself, but also on whether
the evaluation occurs before or aer these moments. Building models
in terms of computational objects with local state forces us to confront
time as an essential concept in programming.
    We can go further in structuring computational models to match our
perception of the physical world. Objects in the world do not change
one at a time in sequence. Rather we perceive them as acting concur-
rently —all at once. So it is oen natural to model systems as collections
of computational processes that execute concurrently. Just as we can
make our programs modular by organizing models in terms of objects
with separate local state, it is oen appropriate to divide computational
models into parts that evolve separately and concurrently. Even if the
programs are to be executed on a sequential computer, the practice of
writing programs as if they were to be executed concurrently forces
the programmer to avoid inessential timing constraints and thus makes
programs more modular.
    In addition to making programs more modular, concurrent compu-
tation can provide a speed advantage over sequential computation. Se-
quential computers execute only one operation at a time, so the amount
of time it takes to perform a task is proportional to the total number
of operations performed.34 However, if it is possible to decompose a
   34 Most real processors actually execute a few operations at a time, following a strat-

egy called pipelining. Although this technique greatly improves the effective utilization
of the hardware, it is used only to speed up the execution of a sequential instruction
stream, while retaining the behavior of the sequential program.

problem into pieces that are relatively independent and need to com-
municate only rarely, it may be possible to allocate pieces to separate
computing processors, producing a speed advantage proportional to the
number of processors available.
    Unfortunately, the complexities introduced by assignment become
even more problematic in the presence of concurrency. e fact of con-
current execution, either because the world operates in parallel or be-
cause our computers do, entails additional complexity in our under-
standing of time.

3.4.1 The Nature of Time in Concurrent Systems
On the surface, time seems straightforward. It is an ordering imposed
on events.35 For any events A and B, either A occurs before B, A and
B are simultaneous, or A occurs aer B. For instance, returning to the
bank account example, suppose that Peter withdraws $10 and Paul with-
draws $25 from a joint account that initially contains $100, leaving $65
in the account. Depending on the order of the two withdrawals, the
sequence of balances in the account is either $100 → $90 → $65 or
$100 → $75 → $65 . In a computer implementation of the banking sys-
tem, this changing sequence of balances could be modeled by successive
assignments to a variable balance.
     In complex situations, however, such a view can be problematic.
Suppose that Peter and Paul, and other people besides, are accessing the
same bank account through a network of banking machines distributed
all over the world. e actual sequence of balances in the account will
depend critically on the detailed timing of the accesses and the details
of the communication among the machines.
  35 Toquote some graffiti seen on a Cambridge building wall: “Time is a device that
was invented to keep everything from happening at once.”

    is indeterminacy in the order of events can pose serious prob-
lems in the design of concurrent systems. For instance, suppose that the
withdrawals made by Peter and Paul are implemented as two separate
processes sharing a common variable balance, each process specified
by the procedure given in Section 3.1.1:
(define (withdraw amount)
  (if (>= balance amount)
         (set! balance (- balance amount))
      "Insufficient funds"))

If the two processes operate independently, then Peter might test the
balance and aempt to withdraw a legitimate amount. However, Paul
might withdraw some funds in between the time that Peter checks the
balance and the time Peter completes the withdrawal, thus invalidating
Peter’s test.
     ings can be worse still. Consider the expression
(set! balance (- balance amount))

executed as part of each withdrawal process. is consists of three steps:
(1) accessing the value of the balance variable; (2) computing the new
balance; (3) seing balance to this new value. If Peter and Paul’s with-
drawals execute this statement concurrently, then the two withdrawals
might interleave the order in which they access balance and set it to
the new value.
    e timing diagram in Figure 3.29 depicts an order of events where
balance starts at 100, Peter withdraws 10, Paul withdraws 25, and yet
the final value of balance is 75. As shown in the diagram, the reason for
this anomaly is that Paul’s assignment of 75 to balance is made under
the assumption that the value of balance to be decremented is 100. at

               Peter                 Bank                Paul


          Access balance: $100
                                               Access balance: $100
         new value: 100 –10 = 90
                                              new value: 100 – 25 = 75
          set! balance to $90


                                               set! balance to $75


       Figure 3.29: Timing diagram showing how interleaving
       the order of events in two banking withdrawals can lead
       to an incorrect final balance.

assumption, however, became invalid when Peter changed balance to
90. is is a catastrophic failure for the banking system, because the
total amount of money in the system is not conserved. Before the trans-
actions, the total amount of money was $100. Aerwards, Peter has $10,
Paul has $25, and the bank has $75.36
    e general phenomenon illustrated here is that several processes
may share a common state variable. What makes this complicated is that
more than one process may be trying to manipulate the shared state at
the same time. For the bank account example, during each transaction,
each customer should be able to act as if the other customers did not
exist. When a customer changes the balance in a way that depends on
the balance, he must be able to assume that, just before the moment of
change, the balance is still what he thought it was.

Correct behavior of concurrent programs
e above example typifies the subtle bugs that can creep into concur-
rent programs. e root of this complexity lies in the assignments to
variables that are shared among the different processes. We already
know that we must be careful in writing programs that use set!, be-
cause the results of a computation depend on the order in which the
     An even worse failure for this system could occur if the two set! operations at-
tempt to change the balance simultaneously, in which case the actual data appearing
in memory might end up being a random combination of the information being writ-
ten by the two processes. Most computers have interlocks on the primitive memory-
write operations, which protect against such simultaneous access. Even this seemingly
simple kind of protection, however, raises implementation challenges in the design of
multiprocessing computers, where elaborate cache-coherence protocols are required to
ensure that the various processors will maintain a consistent view of memory contents,
despite the fact that data may be replicated (“cached”) among the different processors
to increase the speed of memory access.

assignments occur.37 With concurrent processes we must be especially
careful about assignments, because we may not be able to control the
order of the assignments made by the different processes. If several such
changes might be made concurrently (as with two depositors accessing
a joint account) we need some way to ensure that our system behaves
correctly. For example, in the case of withdrawals from a joint bank ac-
count, we must ensure that money is conserved. To make concurrent
programs behave correctly, we may have to place some restrictions on
concurrent execution.
    One possible restriction on concurrency would stipulate that no two
operations that change any shared state variables can occur at the same
time. is is an extremely stringent requirement. For distributed bank-
ing, it would require the system designer to ensure that only one trans-
action could proceed at a time. is would be both inefficient and overly
conservative. Figure 3.30 shows Peter and Paul sharing a bank account,
where Paul has a private account as well. e diagram illustrates two
withdrawals from the shared account (one by Peter and one by Paul)
and a deposit to Paul’s private account.38 e two withdrawals from
the shared account must not be concurrent (since both access and up-
date the same account), and Paul’s deposit and withdrawal must not be
concurrent (since both access and update the amount in Paul’s wallet).
But there should be no problem permiing Paul’s deposit to his pri-
vate account to proceed concurrently with Peter’s withdrawal from the
shared account.
    A less stringent restriction on concurrency would ensure that a con-
    e factorial program in Section 3.1.3 illustrates this for a single sequential process.
    e columns show the contents of Peter’s wallet, the joint account (in Bank1),
Paul’s wallet, and Paul’s private account (in Bank2), before and aer each withdrawal
(W) and deposit (D). Peter withdraws $10 from Bank1; Paul deposits $5 in Bank2, then
withdraws $25 from Bank1.

             Peter          Bank1             Paul         Bank2

              $7             $100             $5            $300

                      W                               D

             $17             $90              $0            $305


             $17             $65              $25           $305


      Figure 3.30: Concurrent deposits and withdrawals from a
      joint account in Bank1 and a private account in Bank2.

current system produces the same result as if the processes had run
sequentially in some order. ere are two important aspects to this re-
quirement. First, it does not require the processes to actually run se-
quentially, but only to produce results that are the same as if they had
run sequentially. For the example in Figure 3.30, the designer of the
bank account system can safely allow Paul’s deposit and Peter’s with-
drawal to happen concurrently, because the net result will be the same
as if the two operations had happened sequentially. Second, there may
be more than one possible “correct” result produced by a concurrent
program, because we require only that the result be the same as for

some sequential order. For example, suppose that Peter and Paul’s joint
account starts out with $100, and Peter deposits $40 while Paul concur-
rently withdraws half the money in the account. en sequential exe-
cution could result in the account balance being either $70 or $90 (see
Exercise 3.38).39
     ere are still weaker requirements for correct execution of con-
current programs. A program for simulating diffusion (say, the flow of
heat in an object) might consist of a large number of processes, each
one representing a small volume of space, that update their values con-
currently. Each process repeatedly changes its value to the average of
its own value and its neighbors’ values. is algorithm converges to the
right answer independent of the order in which the operations are done;
there is no need for any restrictions on concurrent use of the shared val-

       Exercise 3.38: Suppose that Peter, Paul, and Mary share
       a joint bank account that initially contains $100. Concur-
       rently, Peter deposits $10, Paul withdraws $20, and Mary
       withdraws half the money in the account, by executing the
       following commands:
       Peter: (set! balance (+ balance 10))
       Paul:     (set! balance (- balance 20))
       Mary:     (set! balance (- balance (/ balance 2)))

          a. List all the different possible values for balance aer
             these three transactions have been completed, assum-
     A more formal way to express this idea is to say that concurrent programs are
inherently nondeterministic. at is, they are described not by single-valued functions,
but by functions whose results are sets of possible values. In Section 4.3 we will study
a language for expressing nondeterministic computations.

           ing that the banking system forces the three processes
           to run sequentially in some order.
        b. What are some other values that could be produced
           if the system allows the processes to be interleaved?
           Draw timing diagrams like the one in Figure 3.29 to
           explain how these values can occur.

3.4.2 Mechanisms for Controlling Concurrency
We’ve seen that the difficulty in dealing with concurrent processes is
rooted in the need to consider the interleaving of the order of events in
the different processes. For example, suppose we have two processes,
one with three ordered events (a, b, c) and one with three ordered events
(x , y, z). If the two processes run concurrently, with no constraints on
how their execution is interleaved, then there are 20 different possible
orderings for the events that are consistent with the individual order-
ings for the two processes:

(a,b,c,x,y,z)   (a,x,b,y,c,z)    (x,a,b,c,y,z)    (x,a,y,z,b,c)
(a,b,x,c,y,z)   (a,x,b,y,z,c)    (x,a,b,y,c,z)    (x,y,a,b,c,z)
(a,b,x,y,c,z)   (a,x,y,b,c,z)    (x,a,b,y,z,c)    (x,y,a,b,z,c)
(a,b,x,y,z,c)   (a,x,y,b,z,c)    (x,a,y,b,c,z)    (x,y,a,z,b,c)
(a,x,b,c,y,z)   (a,x,y,z,b,c)    (x,a,y,b,z,c)    (x,y,z,a,b,c)

As programmers designing this system, we would have to consider the
effects of each of these 20 orderings and check that each behavior is
acceptable. Such an approach rapidly becomes unwieldy as the numbers
of processes and events increase.
    A more practical approach to the design of concurrent systems is to
devise general mechanisms that allow us to constrain the interleaving of
concurrent processes so that we can be sure that the program behavior

is correct. Many mechanisms have been developed for this purpose. In
this section, we describe one of them, the serializer.

Serializing access to shared state
Serialization implements the following idea: Processes will execute con-
currently, but there will be certain collections of procedures that cannot
be executed concurrently. More precisely, serialization creates distin-
guished sets of procedures such that only one execution of a procedure
in each serialized set is permied to happen at a time. If some procedure
in the set is being executed, then a process that aempts to execute any
procedure in the set will be forced to wait until the first execution has
    We can use serialization to control access to shared variables. For
example, if we want to update a shared variable based on the previ-
ous value of that variable, we put the access to the previous value of
the variable and the assignment of the new value to the variable in the
same procedure. We then ensure that no other procedure that assigns
to the variable can run concurrently with this procedure by serializing
all of these procedures with the same serializer. is guarantees that
the value of the variable cannot be changed between an access and the
corresponding assignment.

Serializers in Scheme
To make the above mechanism more concrete, suppose that we have
extended Scheme to include a procedure called parallel-execute:
(parallel-execute   ⟨p1 ⟩ ⟨p2 ⟩ . . . ⟨pk ⟩)

Each ⟨p⟩ must be a procedure of no arguments. parallel-execute cre-
ates a separate process for each ⟨p⟩, which applies ⟨p⟩ (to no arguments).

ese processes all run concurrently.40
  As an example of how this is used, consider
(define x 10)
 (lambda () (set! x (* x x)))
 (lambda () (set! x (+ x 1))))

is creates two concurrent processes—P1 , which sets x to x times x,
and P2 , which increments x. Aer execution is complete, x will be le
with one of five possible values, depending on the interleaving of the
events of P1 and P2 :
101: P1 sets x to 100 and then P2 increments x to 101.
121: P2 increments x to 11 and then P1 sets x to x * x.
110: P2 changes x from 10 to 11 between the two times that
     P1 accesses the value of x during the evaluation of (* x x).
 11: P2 accesses x, then P1 sets x to 100, then P2 sets x.
100: P1 accesses x (twice), then P2 sets x to 11, then P1 sets x.

We can constrain the concurrency by using serialized procedures, which
are created by serializers. Serializers are constructed by make-serializer,
whose implementation is given below. A serializer takes a procedure as
argument and returns a serialized procedure that behaves like the origi-
nal procedure. All calls to a given serializer return serialized procedures
in the same set.
    us, in contrast to the example above, executing
(define x 10)

  40 parallel-execute   is not part of standard Scheme, but it can be implemented in
 Scheme. In our implementation, the new concurrent processes also run concur-
rently with the original Scheme process. Also, in our implementation, the value re-
turned by parallel-execute is a special control object that can be used to halt the
newly created processes.

(define s (make-serializer))
 (s (lambda () (set! x (* x x))))
 (s (lambda () (set! x (+ x 1)))))

can produce only two possible values for x, 101 or 121. e other pos-
sibilities are eliminated, because the execution of P1 and P2 cannot be
    Here is a version of the make-account procedure from Section 3.1.1,
where the deposits and withdrawals have been serialized:
(define (make-account balance)
  (define (withdraw amount)
    (if (>= balance amount)
        (begin (set! balance (- balance amount))
        "Insufficient funds"))
  (define (deposit amount)
    (set! balance (+ balance amount))
  (let ((protected (make-serializer)))
    (define (dispatch m)
      (cond ((eq? m 'withdraw) (protected withdraw))
             ((eq? m 'deposit) (protected deposit))
             ((eq? m 'balance) balance)
             (else (error "Unknown request: MAKE-ACCOUNT"

With this implementation, two processes cannot be withdrawing from
or depositing into a single account concurrently. is eliminates the
source of the error illustrated in Figure 3.29, where Peter changes the
account balance between the times when Paul accesses the balance to
compute the new value and when Paul actually performs the assign-

ment. On the other hand, each account has its own serializer, so that
deposits and withdrawals for different accounts can proceed concur-

     Exercise 3.39: Which of the five possibilities in the par-
     allel execution shown above remain if we instead serialize
     execution as follows:
     (define x 10)
     (define s (make-serializer))
       (lambda () (set! x ((s (lambda () (* x x))))))
       (s (lambda () (set! x (+ x 1)))))

     Exercise 3.40: Give all possible values of x that can result
     from executing
     (define x 10)
     (parallel-execute (lambda () (set! x (* x x)))
                         (lambda () (set! x (* x x x))))

     Which of these possibilities remain if we instead use seri-
     alized procedures:
     (define x 10)
     (define s (make-serializer))
     (parallel-execute (s (lambda () (set! x (* x x))))
                         (s (lambda () (set! x (* x x x)))))

     Exercise 3.41: Ben Bitdiddle worries that it would be bet-
     ter to implement the bank account as follows (where the
     commented line has been changed):

(define (make-account balance)
  (define (withdraw amount)
    (if (>= balance amount)
        (begin (set! balance
                       (- balance amount))
        "Insufficient funds"))
  (define (deposit amount)
    (set! balance (+ balance amount))
  (let ((protected (make-serializer)))
    (define (dispatch m)
      (cond ((eq? m 'withdraw) (protected withdraw))
             ((eq? m 'deposit) (protected deposit))
             ((eq? m 'balance)
                 (lambda () balance))))   ; serialized
               (error "Unknown request: MAKE-ACCOUNT"

because allowing unserialized access to the bank balance
can result in anomalous behavior. Do you agree? Is there
any scenario that demonstrates Ben’s concern?

Exercise 3.42: Ben Bitdiddle suggests that it’s a waste of
time to create a new serialized procedure in response to
every withdraw and deposit message. He says that make-
account could be changed so that the calls to protected
are done outside the dispatch procedure. at is, an ac-
count would return the same serialized procedure (which

      was created at the same time as the account) each time it is
      asked for a withdrawal procedure.
      (define (make-account balance)
        (define (withdraw amount)
          (if (>= balance amount)
               (begin (set! balance (- balance amount))
               "Insufficient funds"))
        (define (deposit amount)
          (set! balance (+ balance amount))
        (let ((protected (make-serializer)))
          (let ((protected-withdraw (protected withdraw))
                 (protected-deposit (protected deposit)))
             (define (dispatch m)
               (cond ((eq? m 'withdraw) protected-withdraw)
                      ((eq? m 'deposit) protected-deposit)
                      ((eq? m 'balance) balance)
                       (error "Unknown request: MAKE-ACCOUNT"

      Is this a safe change to make? In particular, is there any dif-
      ference in what concurrency is allowed by these two ver-
      sions of make-account?

Complexity of using multiple shared resources
Serializers provide a powerful abstraction that helps isolate the com-
plexities of concurrent programs so that they can be dealt with carefully
and (hopefully) correctly. However, while using serializers is relatively

straightforward when there is only a single shared resource (such as
a single bank account), concurrent programming can be treacherously
difficult when there are multiple shared resources.
    To illustrate one of the difficulties that can arise, suppose we wish to
swap the balances in two bank accounts. We access each account to find
the balance, compute the difference between the balances, withdraw
this difference from one account, and deposit it in the other account.
We could implement this as follows:41
(define (exchange account1 account2)
  (let ((difference (- (account1 'balance)
                              (account2 'balance))))
     ((account1 'withdraw) difference)
     ((account2 'deposit) difference)))

is procedure works well when only a single process is trying to do
the exchange. Suppose, however, that Peter and Paul both have access
to accounts a1, a2, and a3, and that Peter exchanges a1 and a2 while
Paul concurrently exchanges a1 and a3. Even with account deposits and
withdrawals serialized for individual accounts (as in the make-account
procedure shown above in this section), exchange can still produce in-
correct results. For example, Peter might compute the difference in the
balances for a1 and a2, but then Paul might change the balance in a1
before Peter is able to complete the exchange.42 For correct behavior,
we must arrange for the exchange procedure to lock out any other con-
current accesses to the accounts during the entire time of the exchange.
  41 We   have simplified exchange by exploiting the fact that our deposit message ac-
cepts negative amounts. (is is a serious bug in our banking system!)
  42 If the account balances start out as $10, $20, and $30, then aer any number of

concurrent exchanges, the balances should still be $10, $20, and $30 in some order.
Serializing the deposits to individual accounts is not sufficient to guarantee this. See
Exercise 3.43.

    One way we can accomplish this is by using both accounts’ seri-
alizers to serialize the entire exchange procedure. To do this, we will
arrange for access to an account’s serializer. Note that we are deliber-
ately breaking the modularity of the bank-account object by exposing
the serializer. e following version of make-account is identical to the
original version given in Section 3.1.1, except that a serializer is pro-
vided to protect the balance variable, and the serializer is exported via
message passing:
(define (make-account-and-serializer balance)
  (define (withdraw amount)
     (if (>= balance amount)
          (begin (set! balance (- balance amount))
          "Insufficient funds"))
  (define (deposit amount)
     (set! balance (+ balance amount))
  (let ((balance-serializer (make-serializer)))
     (define (dispatch m)
       (cond ((eq? m 'withdraw) withdraw)
                ((eq? m 'deposit) deposit)
                ((eq? m 'balance) balance)
                ((eq? m 'serializer) balance-serializer)
                (else (error "Unknown request: MAKE-ACCOUNT" m))))

We can use this to do serialized deposits and withdrawals. However,
unlike our earlier serialized account, it is now the responsibility of each
user of bank-account objects to explicitly manage the serialization, for
example as follows:43

  43 Exercise  3.45 investigates why deposits and withdrawals are no longer automati-
cally serialized by the account.
(define (deposit account amount)
  (let ((s (account 'serializer))
         (d (account 'deposit)))
    ((s d) amount)))

Exporting the serializer in this way gives us enough flexibility to imple-
ment a serialized exchange program. We simply serialize the original
exchange procedure with the serializers for both accounts:
(define (serialized-exchange account1 account2)
  (let ((serializer1 (account1 'serializer))
         (serializer2 (account2 'serializer)))
    ((serializer1 (serializer2 exchange))

      Exercise 3.43: Suppose that the balances in three accounts
      start out as $10, $20, and $30, and that multiple processes
      run, exchanging the balances in the accounts. Argue that if
      the processes are run sequentially, aer any number of con-
      current exchanges, the account balances should be $10, $20,
      and $30 in some order. Draw a timing diagram like the one
      in Figure 3.29 to show how this condition can be violated
      if the exchanges are implemented using the first version of
      the account-exchange program in this section. On the other
      hand, argue that even with this exchange program, the sum
      of the balances in the accounts will be preserved. Draw a
      timing diagram to show how even this condition would be
      violated if we did not serialize the transactions on individ-
      ual accounts.
      Exercise 3.44: Consider the problem of transferring an amount
      from one account to another. Ben Bitdiddle claims that this

can be accomplished with the following procedure, even if
there are multiple people concurrently transferring money
among multiple accounts, using any account mechanism
that serializes deposit and withdrawal transactions, for ex-
ample, the version of make-account in the text above.
(define (transfer from-account to-account amount)
  ((from-account 'withdraw) amount)
  ((to-account 'deposit) amount))

Louis Reasoner claims that there is a problem here, and
that we need to use a more sophisticated method, such as
the one required for dealing with the exchange problem. Is
Louis right? If not, what is the essential difference between
the transfer problem and the exchange problem? (You should
assume that the balance in from-account is at least amount.)

Exercise 3.45: Louis Reasoner thinks our bank-account sys-
tem is unnecessarily complex and error-prone now that de-
posits and withdrawals aren’t automatically serialized. He
suggests that make-account-and-serializer should have
exported the serializer (for use by such procedures as serialized-
exchange) in addition to (rather than instead o) using it
to serialize accounts and deposits as make-account did. He
proposes to redefine accounts as follows:
(define (make-account-and-serializer balance)
  (define (withdraw amount)
    (if (>= balance amount)
        (begin (set! balance (- balance amount)) balance)
        "Insufficient funds"))
  (define (deposit amount)
    (set! balance (+ balance amount)) balance)

          (let ((balance-serializer (make-serializer)))
            (define (dispatch m)
              (cond ((eq? m 'withdraw) (balance-serializer withdraw))
                      ((eq? m 'deposit) (balance-serializer deposit))
                      ((eq? m 'balance) balance)
                      ((eq? m 'serializer) balance-serializer)
                      (else (error "Unknown request: MAKE-ACCOUNT" m))))

       en deposits are handled as with the original make-account:
       (define (deposit account amount)
          ((account 'deposit) amount))

       Explain what is wrong with Louis’s reasoning. In particu-
       lar, consider what happens when serialized-exchange is

Implementing serializers
We implement serializers in terms of a more primitive synchroniza-
tion mechanism called a mutex. A mutex is an object that supports two
operations—the mutex can be acquired, and the mutex can be released.
Once a mutex has been acquired, no other acquire operations on that
mutex may proceed until the mutex is released.44 In our implementa-
  44 e   term “mutex” is an abbreviation for mutual exclusion. e general problem of
arranging a mechanism that permits concurrent processes to safely share resources is
called the mutual exclusion problem. Our mutex is a simple variant of the semaphore
mechanism (see Exercise 3.47), which was introduced in the “THE” Multiprogramming
System developed at the Technological University of Eindhoven and named for the
university’s initials in Dutch (Dijkstra 1968a). e acquire and release operations were
originally called P and V, from the Dutch words passeren (to pass) and vrijgeven (to
release), in reference to the semaphores used on railroad systems. Dijkstra’s classic
exposition (Dijkstra 1968b) was one of the first to clearly present the issues of concur-

tion, each serializer has an associated mutex. Given a procedure p, the
serializer returns a procedure that acquires the mutex, runs p, and then
releases the mutex. is ensures that only one of the procedures pro-
duced by the serializer can be running at once, which is precisely the
serialization property that we need to guarantee.
(define (make-serializer)
  (let ((mutex (make-mutex)))
     (lambda (p)
       (define (serialized-p . args)
          (mutex 'acquire)
          (let ((val (apply p args)))
             (mutex 'release)

e mutex is a mutable object (here we’ll use a one-element list, which
we’ll refer to as a cell ) that can hold the value true or false. When the
value is false, the mutex is available to be acquired. When the value is
true, the mutex is unavailable, and any process that aempts to acquire
the mutex must wait.
    Our mutex constructor make-mutex begins by initializing the cell
contents to false. To acquire the mutex, we test the cell. If the mutex
is available, we set the cell contents to true and proceed. Otherwise,
we wait in a loop, aempting to acquire over and over again, until we
find that the mutex is available.45 To release the mutex, we set the cell
rency control, and showed how to use semaphores to handle a variety of concurrency
   45 In most time-shared operating systems, processes that are blocked by a mutex do

not waste time “busy-waiting” as above. Instead, the system schedules another process
to run while the first is waiting, and the blocked process is awakened when the mutex
becomes available.

contents to false.
(define (make-mutex)
  (let ((cell (list false)))
    (define (the-mutex m)
       (cond ((eq? m 'acquire)
               (if (test-and-set! cell)
                     (the-mutex 'acquire)))    ; retry
              ((eq? m 'release) (clear! cell))))
(define (clear! cell) (set-car! cell false))

test-and-set!     tests the cell and returns the result of the test. In addi-
tion, if the test was false, test-and-set! sets the cell contents to true
before returning false. We can express this behavior as the following
(define (test-and-set! cell)
  (if (car cell) true (begin (set-car! cell true) false)))

However, this implementation of test-and-set! does not suffice as
it stands. ere is a crucial subtlety here, which is the essential place
where concurrency control enters the system: e test-and-set! op-
eration must be performed atomically. at is, we must guarantee that,
once a process has tested the cell and found it to be false, the cell con-
tents will actually be set to true before any other process can test the
cell. If we do not make this guarantee, then the mutex can fail in a way
similar to the bank-account failure in Figure 3.29. (See Exercise 3.46.)
     e actual implementation of test-and-set! depends on the de-
tails of how our system runs concurrent processes. For example, we
might be executing concurrent processes on a sequential processor us-
ing a time-slicing mechanism that cycles through the processes, permit-
ting each process to run for a short time before interrupting it and mov-

ing on to the next process. In that case, test-and-set! can work by dis-
abling time slicing during the testing and seing.46 Alternatively, mul-
tiprocessing computers provide instructions that support atomic oper-
ations directly in hardware.47

          Exercise 3.46: Suppose that we implement test-and-set!
          using an ordinary procedure as shown in the text, without
          aempting to make the operation atomic. Draw a timing
  46 In   Scheme for a single processor, which uses a time-slicing model, test-and-
set!   can be implemented as follows:
(define (test-and-set! cell)
   (lambda ()
       (if (car cell)
            (begin (set-car! cell true)

without-interrupts      disables time-slicing interrupts while its procedure argument is
being executed.
   47 ere are many variants of such instructions—including test-and-set, test-and-

clear, swap, compare-and-exchange, load-reserve, and store-conditional—whose design
must be carefully matched to the machine’s processor-memory interface. One issue that
arises here is to determine what happens if two processes aempt to acquire the same
resource at exactly the same time by using such an instruction. is requires some
mechanism for making a decision about which process gets control. Such a mechanism
is called an arbiter. Arbiters usually boil down to some sort of hardware device. Un-
fortunately, it is possible to prove that one cannot physically construct a fair arbiter
that works 100% of the time unless one allows the arbiter an arbitrarily long time to
make its decision. e fundamental phenomenon here was originally observed by the
fourteenth-century French philosopher Jean Buridan in his commentary on Aristotle’s
De caelo. Buridan argued that a perfectly rational dog placed between two equally at-
tractive sources of food will starve to death, because it is incapable of deciding which
to go to first.

      diagram like the one in Figure 3.29 to demonstrate how the
      mutex implementation can fail by allowing two processes
      to acquire the mutex at the same time.
      Exercise 3.47: A semaphore (of size n) is a generalization of
      a mutex. Like a mutex, a semaphore supports acquire and
      release operations, but it is more general in that up to n
      processes can acquire it concurrently. Additional processes
      that aempt to acquire the semaphore must wait for release
      operations. Give implementations of semaphores

         a. in terms of mutexes
         b. in terms of atomic test-and-set! operations.

Now that we have seen how to implement serializers, we can see that
account exchanging still has a problem, even with the serialized-
exchange procedure above. Imagine that Peter aempts to exchange
a1 with a2 while Paul concurrently aempts to exchange a2 with a1.
Suppose that Peter’s process reaches the point where it has entered a
serialized procedure protecting a1 and, just aer that, Paul’s process en-
ters a serialized procedure protecting a2. Now Peter cannot proceed (to
enter a serialized procedure protecting a2) until Paul exits the serialized
procedure protecting a2. Similarly, Paul cannot proceed until Peter exits
the serialized procedure protecting a1. Each process is stalled forever,
waiting for the other. is situation is called a deadlock. Deadlock is al-
ways a danger in systems that provide concurrent access to multiple
shared resources.
    One way to avoid the deadlock in this situation is to give each ac-
count a unique identification number and rewrite serialized-exchange

so that a process will always aempt to enter a procedure protecting the
lowest-numbered account first. Although this method works well for
the exchange problem, there are other situations that require more so-
phisticated deadlock-avoidance techniques, or where deadlock cannot
be avoided at all. (See Exercise 3.48 and Exercise 3.49.)48

       Exercise 3.48: Explain in detail why the deadlock-avoidance
       method described above, (i.e., the accounts are numbered,
       and each process aempts to acquire the smaller-numbered
       account first) avoids deadlock in the exchange problem. Re-
       write serialized-exchange to incorporate this idea. (You
       will also need to modify make-account so that each account
       is created with a number, which can be accessed by sending
       an appropriate message.)

       Exercise 3.49: Give a scenario where the deadlock-avoid-
       ance mechanism described above does not work. (Hint: In
       the exchange problem, each process knows in advance which
       accounts it will need to get access to. Consider a situation
       where a process must get access to some shared resources
       before it can know which additional shared resources it will

  48 e  general technique for avoiding deadlock by numbering the shared resources
and acquiring them in order is due to Havender (1968). Situations where deadlock can-
not be avoided require deadlock-recovery methods, which entail having processes “back
out” of the deadlocked state and try again. Deadlock-recovery mechanisms are widely
used in database management systems, a topic that is treated in detail in Gray and
Reuter 1993.

Concurrency, time, and communication
We’ve seen how programming concurrent systems requires controlling
the ordering of events when different processes access shared state, and
we’ve seen how to achieve this control through judicious use of serial-
izers. But the problems of concurrency lie deeper than this, because,
from a fundamental point of view, it’s not always clear what is meant
by “shared state.”
     Mechanisms such as test-and-set! require processes to examine a
global shared flag at arbitrary times. is is problematic and inefficient
to implement in modern high-speed processors, where due to optimiza-
tion techniques such as pipelining and cached memory, the contents
of memory may not be in a consistent state at every instant. In con-
temporary multiprocessing systems, therefore, the serializer paradigm
is being supplanted by new approaches to concurrency control.49
     e problematic aspects of shared state also arise in large, distributed
systems. For instance, imagine a distributed banking system where indi-
vidual branch banks maintain local values for bank balances and period-
ically compare these with values maintained by other branches. In such
a system the value of “the account balance” would be undetermined,
except right aer synchronization. If Peter deposits money in an ac-
count he holds jointly with Paul, when should we say that the account
balance has changed—when the balance in the local branch changes, or
not until aer the synchronization? And if Paul accesses the account
  49 One such alternative to serialization is called barrier synchronization. e program-

mer permits concurrent processes to execute as they please, but establishes certain
synchronization points (“barriers”) through which no process can proceed until all the
processes have reached the barrier. Modern processors provide machine instructions
that permit programmers to establish synchronization points at places where consis-
tency is required. e PowerPC, for example, includes for this purpose two instructions
called  and  (Enforced In-order Execution of Input/Output).

from a different branch, what are the reasonable constraints to place on
the banking system such that the behavior is “correct”? e only thing
that might maer for correctness is the behavior observed by Peter and
Paul individually and the “state” of the account immediately aer syn-
chronization. estions about the “real” account balance or the order of
events between synchronizations may be irrelevant or meaningless.50
    e basic phenomenon here is that synchronizing different pro-
cesses, establishing shared state, or imposing an order on events re-
quires communication among the processes. In essence, any notion of
time in concurrency control must be intimately tied to communica-
tion.51 It is intriguing that a similar connection between time and com-
munication also arises in the eory of Relativity, where the speed of
light (the fastest signal that can be used to synchronize events) is a
fundamental constant relating time and space. e complexities we en-
counter in dealing with time and state in our computational models may
in fact mirror a fundamental complexity of the physical universe.

3.5 Streams
We’ve gained a good understanding of assignment as a tool in modeling,
as well as an appreciation of the complex problems that assignment
raises. It is time to ask whether we could have gone about things in a
different way, so as to avoid some of these problems. In this section,
  50 is  may seem like a strange point of view, but there are systems that work this
way. International charges to credit-card accounts, for example, are normally cleared
on a per-country basis, and the charges made in different countries are periodically
reconciled. us the account balance may be different in different countries.
  51 For distributed systems, this perspective was pursued by Lamport (1978), who

showed how to use communication to establish “global clocks” that can be used to
establish orderings on events in distributed systems.

we explore an alternative approach to modeling state, based on data
structures called streams. As we shall see, streams can mitigate some of
the complexity of modeling state.
     Let’s step back and review where this complexity comes from. In
an aempt to model real-world phenomena, we made some apparently
reasonable decisions: We modeled real-world objects with local state by
computational objects with local variables. We identified time variation
in the real world with time variation in the computer. We implemented
the time variation of the states of the model objects in the computer
with assignments to the local variables of the model objects.
     Is there another approach? Can we avoid identifying time in the
computer with time in the modeled world? Must we make the model
change with time in order to model phenomena in a changing world?
ink about the issue in terms of mathematical functions. We can de-
scribe the time-varying behavior of a quantity x as a function of time
x (t). If we concentrate on x instant by instant, we think of it as a chang-
ing quantity. Yet if we concentrate on the entire time history of values,
we do not emphasize change—the function itself does not change.52
     If time is measured in discrete steps, then we can model a time func-
tion as a (possibly infinite) sequence. In this section, we will see how to
model change in terms of sequences that represent the time histories
of the systems being modeled. To accomplish this, we introduce new
data structures called streams. From an abstract point of view, a stream
is simply a sequence. However, we will find that the straightforward
implementation of streams as lists (as in Section 2.2.1) doesn’t fully re-
  52 Physicists sometimes adopt this view by introducing the “world lines” of particles

as a device for reasoning about motion. We’ve also already mentioned (Section 2.2.3)
that this is the natural way to think about signal-processing systems. We will explore
applications of streams to signal processing in Section 3.5.3.

veal the power of stream processing. As an alternative, we introduce
the technique of delayed evaluation, which enables us to represent very
large (even infinite) sequences as streams.
    Stream processing lets us model systems that have state without
ever using assignment or mutable data. is has important implications,
both theoretical and practical, because we can build models that avoid
the drawbacks inherent in introducing assignment. On the other hand,
the stream framework raises difficulties of its own, and the question
of which modeling technique leads to more modular and more easily
maintained systems remains open.

3.5.1 Streams Are Delayed Lists
As we saw in Section 2.2.3, sequences can serve as standard interfaces
for combining program modules. We formulated powerful abstractions
for manipulating sequences, such as map, filter, and accumulate, that
capture a wide variety of operations in a manner that is both succinct
and elegant.
    Unfortunately, if we represent sequences as lists, this elegance is
bought at the price of severe inefficiency with respect to both the time
and space required by our computations. When we represent manip-
ulations on sequences as transformations of lists, our programs must
construct and copy data structures (which may be huge) at every step
of a process.
    To see why this is true, let us compare two programs for computing
the sum of all the prime numbers in an interval. e first program is
wrien in standard iterative style:53

  53 Assume   that we have a predicate prime? (e.g., as in Section 1.2.6) that tests for

(define (sum-primes a b)
  (define (iter count accum)
    (cond ((> count b) accum)
           ((prime? count)
               (iter (+ count 1) (+ count accum)))
           (else (iter (+ count 1) accum))))
  (iter a 0))

e second program performs the same computation using the sequence
operations of Section 2.2.3:
(define (sum-primes a b)
  (accumulate +
                (filter prime?
                         (enumerate-interval a b))))

In carrying out the computation, the first program needs to store only
the sum being accumulated. In contrast, the filter in the second pro-
gram cannot do any testing until enumerate-interval has constructed
a complete list of the numbers in the interval. e filter generates an-
other list, which in turn is passed to accumulate before being collapsed
to form a sum. Such large intermediate storage is not needed by the first
program, which we can think of as enumerating the interval incremen-
tally, adding each prime to the sum as it is generated.
     e inefficiency in using lists becomes painfully apparent if we use
the sequence paradigm to compute the second prime in the interval
from 10,000 to 1,000,000 by evaluating the expression
(car (cdr (filter prime?
                    (enumerate-interval 10000 1000000))))

is expression does find the second prime, but the computational over-
head is outrageous. We construct a list of almost a million integers, filter

this list by testing each element for primality, and then ignore almost
all of the result. In a more traditional programming style, we would in-
terleave the enumeration and the filtering, and stop when we reached
the second prime.
     Streams are a clever idea that allows one to use sequence manipu-
lations without incurring the costs of manipulating sequences as lists.
With streams we can achieve the best of both worlds: We can formu-
late programs elegantly as sequence manipulations, while aaining the
efficiency of incremental computation. e basic idea is to arrange to
construct a stream only partially, and to pass the partial construction
to the program that consumes the stream. If the consumer aempts to
access a part of the stream that has not yet been constructed, the stream
will automatically construct just enough more of itself to produce the
required part, thus preserving the illusion that the entire stream exists.
In other words, although we will write programs as if we were process-
ing complete sequences, we design our stream implementation to au-
tomatically and transparently interleave the construction of the stream
with its use.
     On the surface, streams are just lists with different names for the
procedures that manipulate them. ere is a constructor, cons-stream,
and two selectors, stream-car and stream-cdr, which satisfy the con-
(stream-car (cons-stream x y)) = x
(stream-cdr (cons-stream x y)) = y

ere is a distinguishable object, the-empty-stream, which cannot be
the result of any cons-stream operation, and which can be identified
with the predicate stream-null?.54 us we can make and use streams,
  54 In the  implementation, the-empty-stream is the same as the empty list '(),

and stream-null? is the same as null?.

in just the same way as we can make and use lists, to represent aggregate
data arranged in a sequence. In particular, we can build stream analogs
of the list operations from Chapter 2, such as list-ref, map, and for-

(define (stream-ref s n)
  (if (= n 0)
        (stream-car s)
        (stream-ref (stream-cdr s) (- n 1))))
(define (stream-map proc s)
  (if (stream-null? s)
        (cons-stream (proc (stream-car s))
                          (stream-map proc (stream-cdr s)))))
(define (stream-for-each proc s)
  (if (stream-null? s)
        (begin (proc (stream-car s))
                 (stream-for-each proc (stream-cdr s)))))

stream-for-each        is useful for viewing streams:
(define (display-stream s)
  (stream-for-each display-line s))
(define (display-line x) (newline) (display x))

To make the stream implementation automatically and transparently
interleave the construction of a stream with its use, we will arrange for
  55 is  should bother you. e fact that we are defining such similar procedures for
streams and lists indicates that we are missing some underlying abstraction. Unfor-
tunately, in order to exploit this abstraction, we will need to exert finer control over
the process of evaluation than we can at present. We will discuss this point further at
the end of Section 3.5.4. In Section 4.2, we’ll develop a framework that unifies lists and

the cdr of a stream to be evaluated when it is accessed by the stream-
cdr procedure rather than when the stream is constructed by cons-
stream. is implementation choice is reminiscent of our discussion of
rational numbers in Section 2.1.2, where we saw that we can choose
to implement rational numbers so that the reduction of numerator and
denominator to lowest terms is performed either at construction time
or at selection time. e two rational-number implementations produce
the same data abstraction, but the choice has an effect on efficiency.
ere is a similar relationship between streams and ordinary lists. As a
data abstraction, streams are the same as lists. e difference is the time
at which the elements are evaluated. With ordinary lists, both the car
and the cdr are evaluated at construction time. With streams, the cdr
is evaluated at selection time.
    Our implementation of streams will be based on a special form called
delay. Evaluating (delay ⟨exp⟩) does not evaluate the expression ⟨exp ⟩,
but rather returns a so-called delayed object, which we can think of as
a “promise” to evaluate ⟨exp ⟩ at some future time. As a companion to
delay, there is a procedure called force that takes a delayed object as
argument and performs the evaluation—in effect, forcing the delay to
fulfill its promise. We will see below how delay and force can be im-
plemented, but first let us use these to construct streams.
    cons-stream is a special form defined so that

(cons-stream   ⟨a⟩ ⟨b⟩)

is equivalent to
(cons   ⟨a⟩ (delay ⟨b⟩))

What this means is that we will construct streams using pairs. How-
ever, rather than placing the value of the rest of the stream into the cdr
of the pair we will put there a promise to compute the rest if it is ever

requested. stream-car and stream-cdr can now be defined as proce-
(define (stream-car stream) (car stream))
(define (stream-cdr stream) (force (cdr stream)))

stream-car    selects the car of the pair; stream-cdr selects the cdr of
the pair and evaluates the delayed expression found there to obtain the
rest of the stream.56

The stream implementation in action
To see how this implementation behaves, let us analyze the “outra-
geous” prime computation we saw above, reformulated in terms of streams:
  (stream-filter prime?
                       10000 1000000))))

We will see that it does indeed work efficiently.
   We begin by calling stream-enumerate-interval with the argu-
ments 10,000 and 1,000,000. stream-enumerate-interval is the stream
analog of enumerate-interval (Section 2.2.3):
(define (stream-enumerate-interval low high)
  (if (> low high)

  56 Although stream-car and stream-cdr can be defined as procedures, cons-stream

must be a special form. If cons-stream were a procedure, then, according to our model
of evaluation, evaluating (cons-stream ⟨a⟩ ⟨b⟩) would automatically cause ⟨b ⟩ to be
evaluated, which is precisely what we do not want to happen. For the same reason,
delay must be a special form, though force can be an ordinary procedure.

          (stream-enumerate-interval (+ low 1) high))))

and thus the result returned by stream-enumerate-interval, formed
by the cons-stream, is57
(cons 10000
       (delay (stream-enumerate-interval 10001 1000000)))

at is, stream-enumerate-interval returns a stream represented as a
pair whose car is 10,000 and whose cdr is a promise to enumerate more
of the interval if so requested. is stream is now filtered for primes,
using the stream analog of the filter procedure (Section 2.2.3):
(define (stream-filter pred stream)
  (cond ((stream-null? stream) the-empty-stream)
          ((pred (stream-car stream))
           (cons-stream (stream-car stream)
                              (stream-cdr stream))))
          (else (stream-filter pred (stream-cdr stream)))))

stream-filter tests the stream-car of the stream (the car of the pair,
which is 10,000). Since this is not prime, stream-filter examines the
stream-cdr of its input stream. e call to stream-cdr forces evaluation
of the delayed stream-enumerate-interval, which now returns
(cons 10001
       (delay (stream-enumerate-interval 10002 1000000)))

  57 e  numbers shown here do not really appear in the delayed expression. What
actually appears is the original expression, in an environment in which the variables
are bound to the appropriate numbers. For example, (+ low 1) with low bound to
10,000 actually appears where 10001 is shown.

stream-filter      now looks at the stream-car of this stream, 10,001,
sees that this is not prime either, forces another stream-cdr, and so on,
until stream-enumerate-interval yields the prime 10,007, whereupon
stream-filter, according to its definition, returns

(cons-stream (stream-car stream)
              (stream-filter pred (stream-cdr stream)))

which in this case is
(cons 10007
      (delay (stream-filter
               (cons 10008
                        (delay (stream-enumerate-interval

is result is now passed to stream-cdr in our original expression. is
forces the delayed stream-filter, which in turn keeps forcing the de-
layed stream-enumerate-interval until it finds the next prime, which
is 10,009. Finally, the result passed to stream-car in our original ex-
pression is
(cons 10009
      (delay (stream-filter
               (cons 10010
                        (delay (stream-enumerate-interval

stream-car returns 10,009, and the computation is complete. Only as
many integers were tested for primality as were necessary to find the

second prime, and the interval was enumerated only as far as was nec-
essary to feed the prime filter.
    In general, we can think of delayed evaluation as “demand-driven”
programming, whereby each stage in the stream process is activated
only enough to satisfy the next stage. What we have done is to decouple
the actual order of events in the computation from the apparent struc-
ture of our procedures. We write procedures as if the streams existed “all
at once” when, in reality, the computation is performed incrementally,
as in traditional programming styles.

Implementing delay and force
Although delay and force may seem like mysterious operations, their
implementation is really quite straightforward. delay must package an
expression so that it can be evaluated later on demand, and we can ac-
complish this simply by treating the expression as the body of a proce-
dure. delay can be a special form such that
(delay   ⟨exp⟩)

is syntactic sugar for
(lambda ()   ⟨exp⟩)

force simply calls the procedure (of no arguments) produced by delay,
so we can implement force as a procedure:
(define (force delayed-object) (delayed-object))

is implementation suffices for delay and force to work as advertised,
but there is an important optimization that we can include. In many ap-
plications, we end up forcing the same delayed object many times. is
can lead to serious inefficiency in recursive programs involving streams.
(See Exercise 3.57.) e solution is to build delayed objects so that the

first time they are forced, they store the value that is computed. Subse-
quent forcings will simply return the stored value without repeating the
computation. In other words, we implement delay as a special-purpose
memoized procedure similar to the one described in Exercise 3.27. One
way to accomplish this is to use the following procedure, which takes as
argument a procedure (of no arguments) and returns a memoized ver-
sion of the procedure. e first time the memoized procedure is run, it
saves the computed result. On subsequent evaluations, it simply returns
the result.
(define (memo-proc proc)
  (let ((already-run? false) (result false))
     (lambda ()
        (if (not already-run?)
             (begin (set! result (proc))
                      (set! already-run? true)

delay   is then defined so that (delay ⟨exp⟩) is equivalent to
(memo-proc (lambda ()        ⟨exp⟩))

and force is as defined previously.58

  58 ere are many possible implementations of streams other than the one described

in this section. Delayed evaluation, which is the key to making streams practical, was
inherent in Algol 60’s call-by-name parameter-passing method. e use of this mech-
anism to implement streams was first described by Landin (1965). Delayed evaluation
for streams was introduced into Lisp by Friedman and Wise (1976). In their implemen-
tation, cons always delays evaluating its arguments, so that lists automatically behave
as streams. e memoizing optimization is also known as call-by-need. e Algol com-
munity would refer to our original delayed objects as call-by-name thunks and to the
optimized versions as call-by-need thunks.

       Exercise 3.50: Complete the following definition, which
       generalizes stream-map to allow procedures that take mul-
       tiple arguments, analogous to map in Section 2.2.1, Footnote
       (define (stream-map proc . argstreams)
          (if (⟨??⟩ (car argstreams))
                  (apply proc (map      ⟨??⟩ argstreams))
                  (apply stream-map
                          (cons proc (map       ⟨??⟩ argstreams))))))

       Exercise 3.51: In order to take a closer look at delayed eval-
       uation, we will use the following procedure, which simply
       returns its argument aer printing it:
       (define (show x)
          (display-line x)

       What does the interpreter print in response to evaluating
       each expression in the following sequence?59
       (define x

  59 Exercises such as Exercise 3.51 and Exercise 3.52 are valuable for testing our un-
derstanding of how delay works. On the other hand, intermixing delayed evaluation
with printing—and, even worse, with assignment—is extremely confusing, and instruc-
tors of courses on computer languages have traditionally tormented their students with
examination questions such as the ones in this section. Needless to say, writing pro-
grams that depend on such subtleties is odious programming style. Part of the power
of stream processing is that it lets us ignore the order in which events actually happen
in our programs. Unfortunately, this is precisely what we cannot afford to do in the
presence of assignment, which forces us to be concerned with time and change.

        (stream-map show
                      (stream-enumerate-interval 0 10)))
      (stream-ref x 5)
      (stream-ref x 7)

      Exercise 3.52: Consider the sequence of expressions
      (define sum 0)
      (define (accum x) (set! sum (+ x sum)) sum)
      (define seq
        (stream-map accum
                      (stream-enumerate-interval 1 20)))
      (define y (stream-filter even? seq))
      (define z
        (stream-filter (lambda (x) (= (remainder x 5) 0))
      (stream-ref y 7)
      (display-stream z)

      What is the value of sum aer each of the above expressions
      is evaluated? What is the printed response to evaluating
      the stream-ref and display-stream expressions? Would
      these responses differ if we had implemented (delay ⟨exp⟩)
      simply as (lambda () ⟨exp⟩) without using the optimiza-
      tion provided by memo-proc? Explain.

3.5.2 Infinite Streams
We have seen how to support the illusion of manipulating streams as
complete entities even though, in actuality, we compute only as much
of the stream as we need to access. We can exploit this technique to rep-
resent sequences efficiently as streams, even if the sequences are very

long. What is more striking, we can use streams to represent sequences
that are infinitely long. For instance, consider the following definition
of the stream of positive integers:
(define (integers-starting-from n)
  (cons-stream n (integers-starting-from (+ n 1))))
(define integers (integers-starting-from 1))

is makes sense because integers will be a pair whose car is 1 and
whose cdr is a promise to produce the integers beginning with 2. is
is an infinitely long stream, but in any given time we can examine only
a finite portion of it. us, our programs will never know that the entire
infinite stream is not there.
    Using integers we can define other infinite streams, such as the
stream of integers that are not divisible by 7:
(define (divisible? x y) (= (remainder x y) 0))
(define no-sevens
  (stream-filter (lambda (x) (not (divisible? x 7)))

en we can find integers not divisible by 7 simply by accessing ele-
ments of this stream:
(stream-ref no-sevens 100)

In analogy with integers, we can define the infinite stream of Fibonacci
(define (fibgen a b) (cons-stream a (fibgen b (+ a b))))
(define fibs (fibgen 0 1))

fibs is a pair whose car is 0 and whose cdr is a promise to evaluate
(fibgen 1 1). When we evaluate this delayed (fibgen 1 1), it will

produce a pair whose car is 1 and whose cdr is a promise to evaluate
(fibgen 1 2), and so on.
    For a look at a more exciting infinite stream, we can generalize the
no-sevens example to construct the infinite stream of prime numbers,
using a method known as the sieve of Eratosthenes.60 We start with the
integers beginning with 2, which is the first prime. To get the rest of
the primes, we start by filtering the multiples of 2 from the rest of the
integers. is leaves a stream beginning with 3, which is the next prime.
Now we filter the multiples of 3 from the rest of this stream. is leaves
a stream beginning with 5, which is the next prime, and so on. In other
words, we construct the primes by a sieving process, described as fol-
lows: To sieve a stream S, form a stream whose first element is the first
element of S and the rest of which is obtained by filtering all multiples
of the first element of S out of the rest of S and sieving the result. is
process is readily described in terms of stream operations:
(define (sieve stream)
    (stream-car stream)
    (sieve (stream-filter
               (lambda (x)
                  (not (divisible? x (stream-car stream))))
               (stream-cdr stream)))))
(define primes (sieve (integers-starting-from 2)))

   60 Eratosthenes, a third-century .. Alexandrian Greek philosopher, is famous for

giving the first accurate estimate of the circumference of the Earth, which he computed
by observing shadows cast at noon on the day of the summer solstice. Eratosthenes’s
sieve method, although ancient, has formed the basis for special-purpose hardware
“sieves” that, until recently, were the most powerful tools in existence for locating large
primes. Since the 70s, however, these methods have been superseded by outgrowths of
the probabilistic techniques discussed in Section 1.2.6.


              cdr        filter:               sieve

       Figure 3.31: e prime sieve viewed as a signal-processing

Now to find a particular prime we need only ask for it:
(stream-ref primes 50)

It is interesting to contemplate the signal-processing system set up by
sieve, shown in the “Henderson diagram” in Figure 3.31.61 e input
stream feeds into an “unconser” that separates the first element of the
stream from the rest of the stream. e first element is used to construct
a divisibility filter, through which the rest is passed, and the output of
the filter is fed to another sieve box. en the original first element is
consed onto the output of the internal sieve to form the output stream.
us, not only is the stream infinite, but the signal processor is also
infinite, because the sieve contains a sieve within it.

  61 We  have named these figures aer Peter Henderson, who was the first person to
show us diagrams of this sort as a way of thinking about stream processing. Each solid
line represents a stream of values being transmied. e dashed line from the car to
the cons and the filter indicates that this is a single value rather than a stream.

Defining streams implicitly
e integers and fibs streams above were defined by specifying “gen-
erating” procedures that explicitly compute the stream elements one by
one. An alternative way to specify streams is to take advantage of de-
layed evaluation to define streams implicitly. For example, the following
expression defines the stream ones to be an infinite stream of ones:
(define ones (cons-stream 1 ones))

is works much like the definition of a recursive procedure: ones is
a pair whose car is 1 and whose cdr is a promise to evaluate ones.
Evaluating the cdr gives us again a 1 and a promise to evaluate ones,
and so on.
    We can do more interesting things by manipulating streams with
operations such as add-streams, which produces the elementwise sum
of two given streams:62
(define (add-streams s1 s2) (stream-map + s1 s2))

Now we can define the integers as follows:
(define integers
  (cons-stream 1 (add-streams ones integers)))

is defines integers to be a stream whose first element is 1 and the rest
of which is the sum of ones and integers. us, the second element of
integers is 1 plus the first element of integers, or 2; the third element
of integers is 1 plus the second element of integers, or 3; and so on.
is definition works because, at any point, enough of the integers
stream has been generated so that we can feed it back into the definition
to produce the next integer.
    We can define the Fibonacci numbers in the same style:
  62 is   uses the generalized version of stream-map from Exercise 3.50.

(define fibs
   (cons-stream 1 (add-streams (stream-cdr fibs) fibs))))

is definition says that fibs is a stream beginning with 0 and 1, such
that the rest of the stream can be generated by adding fibs to itself
shied by one place:
       1   1   2   3   . . . = (stream-cdr fibs)
                       5   8   13   21
       0   1   1   2   . . . = fibs
                       3   5   8    13
0 1 1 2 3 5 8 13 21 34 . . . = fibs

scale-stream  is another useful procedure in formulating such stream
definitions. is multiplies each item in a stream by a given constant:
(define (scale-stream stream factor)
  (stream-map (lambda (x) (* x factor))

For example,
(define double (cons-stream 1 (scale-stream double 2)))

produces the stream of powers of 2: 1, 2, 4, 8, 16, 32, . . ..
    An alternate definition of the stream of primes can be given by start-
ing with the integers and filtering them by testing for primality. We will
need the first prime, 2, to get started:
(define primes
   (stream-filter prime? (integers-starting-from 3))))

is definition is not so straightforward as it appears, because we will
test whether a number n is prime by checking whether n is divisible by
a prime (not by just any integer) less than or equal to n:

(define (prime? n)
   (define (iter ps)
      (cond ((> (square (stream-car ps)) n) true)
               ((divisible? n (stream-car ps)) false)
               (else (iter (stream-cdr ps)))))
   (iter primes))

is is a recursive definition, since primes is defined in terms of the
prime?  predicate, which itself uses the primes stream. e reason this
procedure works is that, at any point, enough of the primes stream has
been generated to test the primality of the numbers we need to check
next. at is, for every n we test for primality, either n is not prime (in
which case there is a prime already generated that divides it) or n is
prime (in which case there is a prime already generated—i.e., a prime
less than n—that is greater than n).63

           Exercise 3.53: Without running the program, describe the
           elements of the stream defined by
           (define s (cons-stream 1 (add-streams s s)))

           Exercise 3.54: Define a procedure mul-streams, analogous
           to add-streams, that produces the elementwise product of
           its two input streams. Use this together with the stream of
           integers to complete the following definition of the stream
           whose n th element (counting from 0) is n + 1 factorial:
  63 is last point is very subtle and relies on the fact that p ≤ pn2 . (Here, p k denotes
the k thprime.) Estimates such as these are very difficult to establish. e ancient proof
by Euclid that there are an infinite number of primes shows that pn+1 ≤ p 1 p 2 · · · pn + 1,
and no substantially beer result was proved until 1851, when the Russian mathemati-
cian P. L. Chebyshev established that pn+1 ≤ 2pn for all n. is result, originally con-
jectured in 1845, is known as Bertrand’s hypothesis. A proof can be found in section 22.3
of Hardy and Wright 1960.

(define factorials
  (cons-stream 1 (mul-streams           ⟨??⟩ ⟨??⟩)))

Exercise 3.55: Define a procedure partial-sums that takes
as argument a stream S and returns the stream whose ele-
ments are S 0 , S 0 +S 1 , S 0 +S 1 +S 2 , . . .. For example, (partial-
sums integers) should be the stream 1, 3, 6, 10, 15, . . ..

Exercise 3.56: A famous problem, first raised by R. Ham-
ming, is to enumerate, in ascending order with no repeti-
tions, all positive integers with no prime factors other than
2, 3, or 5. One obvious way to do this is to simply test each
integer in turn to see whether it has any factors other than
2, 3, and 5. But this is very inefficient, since, as the integers
get larger, fewer and fewer of them fit the requirement. As
an alternative, let us call the required stream of numbers S
and notice the following facts about it.

    • S begins with 1.
    • e elements of (scale-stream S 2) are also ele-
      ments of S.
    • e same is true for (scale-stream S 3) and (scale-
      stream 5 S).

    • ese are all the elements of S.

Now all we have to do is combine elements from these sources.
For this we define a procedure merge that combines two or-
dered streams into one ordered result stream, eliminating

(define (merge s1 s2)
  (cond ((stream-null? s1) s2)
        ((stream-null? s2) s1)
         (let ((s1car (stream-car s1))
                (s2car (stream-car s2)))
           (cond ((< s1car s2car)
                   (merge (stream-cdr s1) s2)))
                 ((> s1car s2car)
                   (merge s1 (stream-cdr s2))))
                   (merge (stream-cdr s1)
                           (stream-cdr s2)))))))))

en the required stream may be constructed with merge,
as follows:
(define S (cons-stream 1 (merge   ⟨??⟩ ⟨??⟩)))

Fill in the missing expressions in the places marked ⟨⁇⟩

Exercise 3.57: How many additions are performed when
we compute the n th Fibonacci number using the definition
of fibs based on the add-streams procedure? Show that
the number of additions would be exponentially greater
if we had implemented (delay ⟨exp⟩) simply as (lambda

       () ⟨exp⟩), without using the optimization provided by the
       memo-proc procedure described in Section

       Exercise 3.58: Give an interpretation of the stream com-
       puted by the following procedure:
       (define (expand num den radix)
            (quotient (* num radix) den)
            (expand (remainder (* num radix) den) den radix)))

       (quotient is a primitive that returns the integer quotient of
       two integers.) What are the successive elements produced
       by (expand 1 7 10)? What is produced by (expand 3 8

       Exercise 3.59: In Section 2.5.3 we saw how to implement
       a polynomial arithmetic system representing polynomials
       as lists of terms. In a similar way, we can work with power
       series, such as

                             x2   x3   x4
                   e = 1+x +
                                +    +    +...,
                             2 3·2 4·3·2
                                        x2   x4
                          cos x = 1 −      +    −...,
                                        2 4·3·2
                                       x3          x5
                         sin x = x −         +             −...
                                       3·2       5·4·3·2
   64 is exercise shows how call-by-need is closely related to ordinary memoization as

described in Exercise 3.27. In that exercise, we used assignment to explicitly construct
a local table. Our call-by-need stream optimization effectively constructs such a table
automatically, storing values in the previously forced parts of the stream.

represented as infinite streams. We will represent the series
a 0 + a 1x + a 2x 2 + a 3x 3 + . . . as the stream whose elements
are the coefficients a 0 , a 1 , a 2 , a 3 , . . ..

  a. e integral of the series a 0 + a 1x + a 2x 2 + a 3x 3 + . . .
     is the series
                        1        1        1
              c + a 0x + a 1x 2 + a 2x 3 + a 3x 4 + . . . ,
                        2        3        4
      where c is any constant. Define a procedure integrate-
      series that takes as input a stream a 0 , a 1 , a 2 , . . . rep-
      resenting a power series and returns the stream a 0 ,
        a , 1 a , . . . of coefficients of the non-constant terms
      2 1 3 2
      of the integral of the series. (Since the result has no
      constant term, it doesn’t represent a power series; when
      we use integrate-series, we will cons on the ap-
      propriate constant.)
  b. e function x 7→ e x is its own derivative. is im-
     plies that e x and the integral of e x are the same se-
     ries, except for the constant term, which is e 0 = 1.
     Accordingly, we can generate the series for e x as
      (define exp-series
        (cons-stream 1 (integrate-series exp-series)))

      Show how to generate the series for sine and cosine,
      starting from the facts that the derivative of sine is
      cosine and the derivative of cosine is the negative of
      (define cosine-series (cons-stream 1          ⟨??⟩))
      (define sine-series (cons-stream 0          ⟨??⟩))

Exercise 3.60: With power series represented as streams
of coefficients as in Exercise 3.59, adding series is imple-
mented by add-streams. Complete the definition of the fol-
lowing procedure for multiplying series:
(define (mul-series s1 s2)
  (cons-stream   ⟨??⟩ (add-streams ⟨??⟩ ⟨??⟩)))

You can test your procedure by verifying that sin2 x + cos2 x = 1,
using the series from Exercise 3.59.

Exercise 3.61: Let S be a power series (Exercise 3.59) whose
constant term is 1. Suppose we want to find the power se-
ries 1/S, that is, the series X such that S X = 1. Write
S = 1 + S R where S R is the part of S aer the constant
term. en we can solve for X as follows:
                         S·X      = 1,
                 (1 + S R ) · X   = 1,
                  X + SR · X      = 1,
                              X   = 1 − SR · X .

In other words, X is the power series whose constant term
is 1 and whose higher-order terms are given by the negative
of S R times X . Use this idea to write a procedure invert-
unit-series that computes 1/S for a power series S with
constant term 1. You will need to use mul-series from Ex-
ercise 3.60.

Exercise 3.62: Use the results of Exercise 3.60 and Exer-
cise 3.61 to define a procedure div-series that divides two
power series. div-series should work for any two series,

      provided that the denominator series begins with a nonzero
      constant term. (If the denominator has a zero constant term,
      then div-series should signal an error.) Show how to use
      div-series together with the result of Exercise 3.59 to gen-
      erate the power series for tangent.

3.5.3 Exploiting the Stream Paradigm
Streams with delayed evaluation can be a powerful modeling tool, pro-
viding many of the benefits of local state and assignment. Moreover,
they avoid some of the theoretical tangles that accompany the intro-
duction of assignment into a programming language.
    e stream approach can be illuminating because it allows us to
build systems with different module boundaries than systems organized
around assignment to state variables. For example, we can think of an
entire time series (or signal) as a focus of interest, rather than the values
of the state variables at individual moments. is makes it convenient
to combine and compare components of state from different moments.

Formulating iterations as stream processes
In Section 1.2.1, we introduced iterative processes, which proceed by
updating state variables. We know now that we can represent state as
a “timeless” stream of values rather than as a set of variables to be up-
dated. Let’s adopt this perspective in revisiting the square-root proce-
dure from Section 1.1.7. Recall that the idea is to generate a sequence of
beer and beer guesses for the square root of x by applying over and
over again the procedure that improves guesses:
(define (sqrt-improve guess x)
  (average guess (/ x guess)))

In our original sqrt procedure, we made these guesses be the successive
values of a state variable. Instead we can generate the infinite stream of
guesses, starting with an initial guess of 1:65
(define (sqrt-stream x)
  (define guesses
      (stream-map (lambda (guess) (sqrt-improve guess x))

(display-stream (sqrt-stream 2))

We can generate more and more terms of the stream to get beer and
beer guesses. If we like, we can write a procedure that keeps generating
terms until the answer is good enough. (See Exercise 3.64.)
    Another iteration that we can treat in the same way is to generate
an approximation to π , based upon the alternating series that we saw
in Section 1.3.1:
                        π         1 1 1
                           = 1− + − +....
                        4         3 5 7
We first generate the stream of summands of the series (the reciprocals
of the odd integers, with alternating signs). en we take the stream of
  65 We can’t use let to bind the local variable guesses, because the value of guesses

depends on guesses itself. Exercise 3.63 addresses why we want a local variable here.

sums of more and more terms (using the partial-sums procedure of
Exercise 3.55) and scale the result by 4:
(define (pi-summands n)
  (cons-stream (/ 1.0 n)
                 (stream-map - (pi-summands (+ n 2)))))
(define pi-stream
  (scale-stream (partial-sums (pi-summands 1)) 4))

(display-stream pi-stream)
is gives us a stream of beer and beer approximations to π , although
the approximations converge rather slowly. Eight terms of the sequence
bound the value of π between 3.284 and 3.017.
     So far, our use of the stream of states approach is not much different
from updating state variables. But streams give us an opportunity to do
some interesting tricks. For example, we can transform a stream with
a sequence accelerator that converts a sequence of approximations to a
new sequence that converges to the same value as the original, only
     One such accelerator, due to the eighteenth-century Swiss math-
ematician Leonhard Euler, works well with sequences that are partial
sums of alternating series (series of terms with alternating signs). In Eu-
ler’s technique, if Sn is the n th term of the original sum sequence, then

the accelerated sequence has terms

                                  (Sn+1 − Sn )2
                      Sn+1 −                      .
                               Sn −1 − 2Sn + Sn+1
us, if the original sequence is represented as a stream of values, the
transformed sequence is given by
(define (euler-transform s)
  (let ((s0 (stream-ref s 0))            ; Sn −1
         (s1 (stream-ref s 1))           ; Sn
         (s2 (stream-ref s 2)))          ; Sn+1
      (cons-stream (- s2 (/ (square (- s2 s1))
                            (+ s0 (* -2 s1) s2)))
                   (euler-transform (stream-cdr s)))))

We can demonstrate Euler acceleration with our sequence of approxi-
mations to π :
(display-stream (euler-transform pi-stream))

Even beer, we can accelerate the accelerated sequence, and recursively
accelerate that, and so on. Namely, we create a stream of streams (a
structure we’ll call a tableau) in which each stream is the transform of
the preceding one:

(define (make-tableau transform s)
  (cons-stream s (make-tableau transform (transform s))))

e tableau has the form

                      s 00 s 01 s 02 s 03 s 04 . . .
                           s 10 s 11 s 12 s 13 . . .
                                s 20 s 21 s 22 . . .

Finally, we form a sequence by taking the first term in each row of the
(define (accelerated-sequence transform s)
  (stream-map stream-car (make-tableau transform s)))

We can demonstrate this kind of “super-acceleration” of the π sequence:
 (accelerated-sequence euler-transform pi-stream))

e result is impressive. Taking eight terms of the sequence yields the
correct value of π to 14 decimal places. If we had used only the original
π sequence, we would need to compute on the order of 1013 terms (i.e.,
expanding the series far enough so that the individual terms are less
than 10 −13 ) to get that much accuracy!

    We could have implemented these acceleration techniques without
using streams. But the stream formulation is particularly elegant and
convenient because the entire sequence of states is available to us as a
data structure that can be manipulated with a uniform set of operations.

      Exercise 3.63: Louis Reasoner asks why the sqrt-stream
      procedure was not wrien in the following more straight-
      forward way, without the local variable guesses:
      (define (sqrt-stream x)
        (cons-stream 1.0 (stream-map
                            (lambda (guess)
                              (sqrt-improve guess x))
                            (sqrt-stream x))))

      Alyssa P. Hacker replies that this version of the procedure
      is considerably less efficient because it performs redundant
      computation. Explain Alyssa’s answer. Would the two ver-
      sions still differ in efficiency if our implementation of delay
      used only (lambda () ⟨exp⟩) without using the optimiza-
      tion provided by memo-proc (Section 3.5.1)?

      Exercise 3.64: Write a procedure stream-limit that takes
      as arguments a stream and a number (the tolerance). It should
      examine the stream until it finds two successive elements
      that differ in absolute value by less than the tolerance, and
      return the second of the two elements. Using this, we could
      compute square roots up to a given tolerance by
      (define (sqrt x tolerance)
        (stream-limit (sqrt-stream x) tolerance))

          Exercise 3.65: Use the series
                                     1 1 1
                          ln 2 = 1 − + − + . . .
                                     2 3 4
          to compute three sequences of approximations to the nat-
          ural logarithm of 2, in the same way we did above for π .
          How rapidly do these sequences converge?

Infinite streams of pairs
In Section 2.2.3, we saw how the sequence paradigm handles traditional
nested loops as processes defined on sequences of pairs. If we generalize
this technique to infinite streams, then we can write programs that are
not easily represented as loops, because the “looping” must range over
an infinite set.
     For example, suppose we want to generalize the prime-sum-pairs
procedure of Section 2.2.3 to produce the stream of pairs of all integers
(i, j) with i ≤ j such that i + j is prime. If int-pairs is the sequence of
all pairs of integers (i, j) with i ≤ j, then our required stream is simply66
 (lambda (pair) (prime? (+ (car pair) (cadr pair))))

Our problem, then, is to produce the stream int-pairs. More generally,
suppose we have two streams S = (Si ) and T = (T j ), and imagine the
infinite rectangular array
                            (S 0 , T0 ) (S 0 , T1 ) (S 0 , T2 ) . . .
                            (S 1 , T0 ) (S 1 , T1 ) (S 1 , T2 ) . . .
                            (S 2 , T0 ) (S 2 , T1 ) (S 2 , T2 ) . . .
  66 As   in Section 2.2.3, we represent a pair of integers as a list rather than a Lisp pair.

We wish to generate a stream that contains all the pairs in the array
that lie on or above the diagonal, i.e., the pairs

                           (S 0 , T0 ) (S 0 , T1 ) (S 0 , T2 ) . . .
                                       (S 1 , T1 ) (S 1 , T2 ) . . .
                                                   (S 2 , T2 ) . . .

(If we take both S and T to be the stream of integers, then this will be
our desired stream int-pairs.)
    Call the general stream of pairs (pairs S T), and consider it to be
composed of three parts: the pair (S 0 , T0 ), the rest of the pairs in the first
row, and the remaining pairs:67

                          (S 0 , T0 ) (S 0 , T1 ) (S 0 , T2 ) . . .
                                      (S 1 , T1 ) (S 1 , T2 ) . . .
                                                  (S 2 , T2 ) . . .

Observe that the third piece in this decomposition (pairs that are not
in the first row) is (recursively) the pairs formed from (stream-cdr S)
and (stream-cdr T). Also note that the second piece (the rest of the
first row) is
(stream-map (lambda (x) (list (stream-car s) x))
                  (stream-cdr t))

us we can form our stream of pairs as follows:
(define (pairs s t)
   (list (stream-car s) (stream-car t))

  67 See   Exercise 3.68 for some insight into why we chose this decomposition.

      (stream-map (lambda (x) (list (stream-car s) x))
                      (stream-cdr t))
      (pairs (stream-cdr s) (stream-cdr t)))))

In order to complete the procedure, we must choose some way to com-
bine the two inner streams. One idea is to use the stream analog of the
append procedure from Section 2.2.1:

(define (stream-append s1 s2)
  (if (stream-null? s1)
       (cons-stream (stream-car s1)
                        (stream-append (stream-cdr s1) s2))))

is is unsuitable for infinite streams, however, because it takes all the
elements from the first stream before incorporating the second stream.
In particular, if we try to generate all pairs of positive integers using
(pairs integers integers)

our stream of results will first try to run through all pairs with the first
integer equal to 1, and hence will never produce pairs with any other
value of the first integer.
     To handle infinite streams, we need to devise an order of combina-
tion that ensures that every element will eventually be reached if we
let our program run long enough. An elegant way to accomplish this is
with the following interleave procedure:68
  68 e  precise statement of the required property on the order of combination is as
follows: ere should be a function f of two arguments such that the pair correspond-
ing to element i of the first stream and element j of the second stream will appear as
element number f (i, j) of the output stream. e trick of using interleave to accom-
plish this was shown to us by David Turner, who employed it in the language KRC
(Turner 1981).

(define (interleave s1 s2)
  (if (stream-null? s1)
      (cons-stream (stream-car s1)
                    (interleave s2 (stream-cdr s1)))))

Since interleave takes elements alternately from the two streams, ev-
ery element of the second stream will eventually find its way into the
interleaved stream, even if the first stream is infinite.
    We can thus generate the required stream of pairs as
(define (pairs s t)
   (list (stream-car s) (stream-car t))
    (stream-map (lambda (x) (list (stream-car s) x))
                 (stream-cdr t))
    (pairs (stream-cdr s) (stream-cdr t)))))

     Exercise 3.66: Examine the stream (pairs integers integers).
     Can you make any general comments about the order in
     which the pairs are placed into the stream? For example,
     approximately how many pairs precede the pair (1, 100)?
     the pair (99, 100)? the pair (100, 100)? (If you can make pre-
     cise mathematical statements here, all the beer. But feel
     free to give more qualitative answers if you find yourself
     geing bogged down.)

     Exercise 3.67: Modify the pairs procedure so that (pairs
     integers integers) will produce the stream of all pairs of
     integers (i, j) (without the condition i ≤ j). Hint: You will
     need to mix in an additional stream.

Exercise 3.68: Louis Reasoner thinks that building a stream
of pairs from three parts is unnecessarily complicated. In-
stead of separating the pair (S 0 , T0 ) from the rest of the pairs
in the first row, he proposes to work with the whole first
row, as follows:
(define (pairs s t)
    (stream-map (lambda (x) (list (stream-car s) x))
    (pairs (stream-cdr s) (stream-cdr t))))

Does this work? Consider what happens if we evaluate (pairs
integers integers) using Louis’s definition of pairs.

Exercise 3.69: Write a procedure triples that takes three
infinite streams, S, T , and U , and produces the stream of
triples (Si , T j , Uk ) such that i ≤ j ≤ k. Use triples to gen-
erate the stream of all Pythagorean triples of positive inte-
gers, i.e., the triples (i, j, k) such that i ≤ j and i 2 + j 2 = k 2 .

Exercise 3.70: It would be nice to be able to generate streams
in which the pairs appear in some useful order, rather than
in the order that results from an ad hoc interleaving pro-
cess. We can use a technique similar to the merge procedure
of Exercise 3.56, if we define a way to say that one pair of
integers is “less than” another. One way to do this is to de-
fine a “weighting function” W (i, j) and stipulate that (i 1 , j 1 )
is less than (i 2 , j 2 ) if W (i 1 , j 1 ) < W (i 2 , j 2 ). Write a proce-
dure merge-weighted that is like merge, except that merge-
weighted takes an additional argument weight, which is a
procedure that computes the weight of a pair, and is used

       to determine the order in which elements should appear in
       the resulting merged stream.69 Using this, generalize pairs
       to a procedure weighted-pairs that takes two streams, to-
       gether with a procedure that computes a weighting func-
       tion, and generates the stream of pairs, ordered according
       to weight. Use your procedure to generate

          a. the stream of all pairs of positive integers (i, j) with
             i ≤ j ordered according to the sum i + j,
          b. the stream of all pairs of positive integers (i, j) with
             i ≤ j, where neither i nor j is divisible by 2, 3, or 5, and
             the pairs are ordered according to the sum 2i + 3j + 5ij.

       Exercise 3.71: Numbers that can be expressed as the sum of
       two cubes in more than one way are sometimes called Ra-
       manujan numbers, in honor of the mathematician Srinivasa
       Ramanujan.70 Ordered streams of pairs provide an elegant
       solution to the problem of computing these numbers. To
       find a number that can be wrien as the sum of two cubes
       in two different ways, we need only generate the stream of
       pairs of integers (i, j) weighted according to the sum i 3 + j 3
  69 We   will require that the weighting function be such that the weight of a pair in-
creases as we move out along a row or down along a column of the array of pairs.
   70 To quote from G. H. Hardy’s obituary of Ramanujan (Hardy 1921): “It was Mr.

Lilewood (I believe) who remarked that ‘every positive integer was one of his friends.’
I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-
cab No. 1729, and remarked that the number seemed to me a rather dull one, and that I
hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number;
it is the smallest number expressible as the sum of two cubes in two different ways.’ ”
e trick of using weighted pairs to generate the Ramanujan numbers was shown to
us by Charles Leiserson.

      (see Exercise 3.70), then search the stream for two consecu-
      tive pairs with the same weight. Write a procedure to gener-
      ate the Ramanujan numbers. e first such number is 1,729.
      What are the next five?

      Exercise 3.72: In a similar way to Exercise 3.71 generate a
      stream of all numbers that can be wrien as the sum of two
      squares in three different ways (showing how they can be
      so wrien).

Streams as signals
We began our discussion of streams by describing them as computa-
tional analogs of the “signals” in signal-processing systems. In fact, we
can use streams to model signal-processing systems in a very direct
way, representing the values of a signal at successive time intervals as
consecutive elements of a stream. For instance, we can implement an
integrator or summer that, for an input stream x = (xi ), an initial value
C, and a small increment dt, accumulates the sum
                            Si = C +          x j dt
                                       j =1

and returns the stream of values S = (Si ). e following integral pro-
cedure is reminiscent of the “implicit style” definition of the stream of
integers (Section 3.5.2):
(define (integral integrand initial-value dt)
  (define int
    (cons-stream initial-value
                   (add-streams (scale-stream integrand dt)


   input                                                   integral
            scale: dt                            cons

      Figure 3.32: e integral procedure viewed as a signal-
      processing system.

Figure 3.32 is a picture of a signal-processing system that corresponds
to the integral procedure. e input stream is scaled by dt and passed
through an adder, whose output is passed back through the same adder.
e self-reference in the definition of int is reflected in the figure by
the feedback loop that connects the output of the adder to one of the

      Exercise 3.73: We can model electrical circuits using streams
      to represent the values of currents or voltages at a sequence
      of times. For instance, suppose we have an RC circuit con-
      sisting of a resistor of resistance R and a capacitor of capac-
      itance C in series. e voltage response v of the circuit to
      an injected current i is determined by the formula in Fig-
      ure 3.33, whose structure is shown by the accompanying
      signal-flow diagram.
      Write a procedure RC that models this circuit. RC should
      take as inputs the values of R, C, and dt and should return
      a procedure that takes as inputs a stream representing the
      current i and an initial value for the capacitor voltage v 0

                                         +           v                      --

            scale: R                                                C

    i              1
            scale:              integral

                                                          Z    t
                                             v = v0 + 1        i dt + R i
                                    v0                C    0

Figure 3.33: An RC circuit and the associated signal-flow diagram.

    and produces as output the stream of voltages v. For ex-
    ample, you should be able to use RC to model an RC circuit
    with R = 5 ohms, C = 1 farad, and a 0.5-second time step by
    evaluating (define RC1 (RC 5 1 0.5)). is defines RC1
    as a procedure that takes a stream representing the time
    sequence of currents and an initial capacitor voltage and
    produces the output stream of voltages.

    Exercise 3.74: Alyssa P. Hacker is designing a system to
    process signals coming from physical sensors. One impor-
    tant feature she wishes to produce is a signal that describes
    the zero crossings of the input signal. at is, the resulting
    signal should be +1 whenever the input signal changes from
    negative to positive, −1 whenever the input signal changes
    from positive to negative, and 0 otherwise. (Assume that
    the sign of a 0 input is positive.) For example, a typical in-

put signal with its associated zero-crossing signal would be
. . . 1 2 1.5 1 0.5 -0.1 -2 -3 -2 -0.5 0.2 3 4 . . .
. . . 0 0 0 0 0 -1 0 0 0 0 1 0 0 . . .

In Alyssa’s system, the signal from the sensor is represented
as a stream sense-data and the stream zero-crossings
is the corresponding stream of zero crossings. Alyssa first
writes a procedure sign-change-detector that takes two
values as arguments and compares the signs of the values
to produce an appropriate 0, 1, or - 1. She then constructs
her zero-crossing stream as follows:
(define (make-zero-crossings input-stream last-value)
    (stream-car input-stream)
    (stream-cdr input-stream)
    (stream-car input-stream))))
(define zero-crossings
  (make-zero-crossings sense-data 0))

Alyssa’s boss, Eva Lu Ator, walks by and suggests that this
program is approximately equivalent to the following one,
which uses the generalized version of stream-map from Ex-
ercise 3.50:
(define zero-crossings
  (stream-map sign-change-detector

Complete the program by supplying the indicated ⟨expression⟩.

Exercise 3.75: Unfortunately, Alyssa’s zero-crossing de-
tector in Exercise 3.74 proves to be insufficient, because the
noisy signal from the sensor leads to spurious zero cross-
ings. Lem E. Tweakit, a hardware specialist, suggests that
Alyssa smooth the signal to filter out the noise before ex-
tracting the zero crossings. Alyssa takes his advice and de-
cides to extract the zero crossings from the signal constructed
by averaging each value of the sense data with the previous
value. She explains the problem to her assistant, Louis Rea-
soner, who aempts to implement the idea, altering Alyssa’s
program as follows:
(define (make-zero-crossings input-stream last-value)
  (let ((avpt (/ (+ (stream-car input-stream)
     (sign-change-detector avpt last-value)
      (stream-cdr input-stream) avpt))))

is does not correctly implement Alyssa’s plan. Find the
bug that Louis has installed and fix it without changing the
structure of the program. (Hint: You will need to increase
the number of arguments to make-zero-crossings.)

Exercise 3.76: Eva Lu Ator has a criticism of Louis’s ap-
proach in Exercise 3.75. e program he wrote is not mod-
ular, because it intermixes the operation of smoothing with
the zero-crossing extraction. For example, the extractor should

      not have to be changed if Alyssa finds a beer way to con-
      dition her input signal. Help Louis by writing a procedure
      smooth that takes a stream as input and produces a stream
      in which each element is the average of two successive in-
      put stream elements. en use smooth as a component to
      implement the zero-crossing detector in a more modular

3.5.4 Streams and Delayed Evaluation
e integral procedure at the end of the preceding section shows how
we can use streams to model signal-processing systems that contain
feedback loops. e feedback loop for the adder shown in Figure 3.32
is modeled by the fact that integral’s internal stream int is defined in
terms of itself:
(define int
   (add-streams (scale-stream integrand dt)

e interpreter’s ability to deal with such an implicit definition depends
on the delay that is incorporated into cons-stream. Without this delay,
the interpreter could not construct int before evaluating both argu-
ments to cons-stream, which would require that int already be defined.
In general, delay is crucial for using streams to model signal-processing
systems that contain loops. Without delay, our models would have to
be formulated so that the inputs to any signal-processing component
would be fully evaluated before the output could be produced. is
would outlaw loops.


                                 dy                    y
                     map: f             integral

       Figure 3.34: An “analog computer circuit” that solves the
       equation dy/dt = f (y).

    Unfortunately, stream models of systems with loops may require
uses of delay beyond the “hidden” delay supplied by cons-stream. For
instance, Figure 3.34 shows a signal-processing system for solving the
differential equation dy/dt = f (y) where f is a given function. e fig-
ure shows a mapping component, which applies f to its input signal,
linked in a feedback loop to an integrator in a manner very similar to
that of the analog computer circuits that are actually used to solve such
    Assuming we are given an initial value y0 for y, we could try to
model this system using the procedure
(define (solve f y0 dt)
  (define y (integral dy y0 dt))
  (define dy (stream-map f y))

is procedure does not work, because in the first line of solve the
call to integral requires that the input dy be defined, which does not
happen until the second line of solve.
    On the other hand, the intent of our definition does make sense,
because we can, in principle, begin to generate the y stream without

knowing dy. Indeed, integral and many other stream operations have
properties similar to those of cons-stream, in that we can generate
part of the answer given only partial information about the arguments.
For integral, the first element of the output stream is the specified
initial-value. us, we can generate the first element of the output
stream without evaluating the integrand dy. Once we know the first
element of y, the stream-map in the second line of solve can begin
working to generate the first element of dy, which will produce the
next element of y, and so on.
    To take advantage of this idea, we will redefine integral to expect
the integrand stream to be a delayed argument. integral will force the
integrand to be evaluated only when it is required to generate more than
the first element of the output stream:
(define (integral delayed-integrand initial-value dt)
  (define int
       (let ((integrand (force delayed-integrand)))
         (add-streams (scale-stream integrand dt) int))))

Now we can implement our solve procedure by delaying the evaluation
of dy in the definition of y:71
(define (solve f y0 dt)
  (define y (integral (delay dy) y0 dt))
  (define dy (stream-map f y))

  71 is procedure is not guaranteed to work in all Scheme implementations, although

for any implementation there is a simple variation that will work. e problem has to
do with subtle differences in the ways that Scheme implementations handle internal
definitions. (See Section 4.1.6.)

In general, every caller of integral must now delay the integrand ar-
gument. We can demonstrate that the solve procedure works by ap-
proximating e ≈ 2.718 by computing the value at y = 1 of the solution
to the differential equation dy/dt = y with initial condition y(0) = 1:
(stream-ref (solve (lambda (y) y)

      Exercise 3.77: e integral procedure used above was
      analogous to the “implicit” definition of the infinite stream
      of integers in Section 3.5.2. Alternatively, we can give a def-
      inition of integral that is more like integers-starting-
      from (also in Section 3.5.2):
      (define (integral integrand initial-value dt)
           (if (stream-null? integrand)
               (integral (stream-cdr integrand)
                           (+ (* dt (stream-car integrand))

      When used in systems with loops, this procedure has the
      same problem as does our original version of integral.
      Modify the procedure so that it expects the integrand as
      a delayed argument and hence can be used in the solve
      procedure shown above.

                         dy0                  y0

       ddy                           dy                    y
                      integral             integral

                      scale: a


                      scale: b

Figure 3.35: Signal-flow diagram for the solution to a
second-order linear differential equation.

Exercise 3.78: Consider the problem of designing a signal-
processing system to study the homogeneous second-order
linear differential equation

                    d 2y    dy
                         − a − by = 0.
                    dt      dt
e output stream, modeling y, is generated by a network
that contains a loop. is is because the value of d 2y/dt 2 de-
pends upon the values of y and dy/dt and both of these are
determined by integrating d 2y/dt 2 . e diagram we would
like to encode is shown in Figure 3.35. Write a procedure
solve-2nd that takes as arguments the constants a, b, and
dt and the initial values y0 and dy0 for y and dy/dt and gen-

                                 +   vR    --

                            iR                       iL
           +       iC                                     +
          vC            C                        L        vL
          --                                              --

               Figure 3.36: A series RLC circuit.

erates the stream of successive values of y.

Exercise 3.79: Generalize the solve-2nd procedure of Ex-
ercise 3.78 so that it can be used to solve general second-
order differential equations d 2y/dt 2 = f (dy/dt , y).

Exercise 3.80: A series RLC circuit consists of a resistor, a
capacitor, and an inductor connected in series, as shown
in Figure 3.36. If R, L, and C are the resistance, inductance,
and capacitance, then the relations between voltage (v) and
current (i) for the three components are described by the
                                      di L                     dvC
    v R = i R R,            vL = L         ,          iC = C       ,
                                      dt                        dt
and the circuit connections dictate the relations

           i R = i L = −i C ,              vC = v L + v R .

Combining these equations shows that the state of the cir-
cuit (summarized by vC , the voltage across the capacitor,

                                scale: 1/L


                      dvC                 v C0


                         diL                       iL
             add                 integral



Figure 3.37: A signal-flow diagram for the solution to a
series RLC circuit.

and i L , the current in the inductor) is described by the pair
of differential equations
          dvC       iL        di L   1    R
               =− ,                = vC − i L .
           dt       C          dt    L    L
e signal-flow diagram representing this system of differ-
ential equations is shown in Figure 3.37.

Write a procedure RLC that takes as arguments the param-
eters R, L, and C of the circuit and the time increment dt.

       In a manner similar to that of the RC procedure of Exercise
       3.73, RLC should produce a procedure that takes the initial
       values of the state variables, vC 0 and i L 0 , and produces a
       pair (using cons) of the streams of states vC and i L . Using
       RLC, generate the pair of streams that models the behavior
       of a series RLC circuit with R = 1 ohm, C = 0.2 farad, L = 1
       henry, dt = 0.1 second, and initial values i L 0 = 0 amps and
       vC 0 = 10 volts.

Normal-order evaluation
e examples in this section illustrate how the explicit use of delay
and force provides great programming flexibility, but the same exam-
ples also show how this can make our programs more complex. Our new
integral procedure, for instance, gives us the power to model systems
with loops, but we must now remember that integral should be called
with a delayed integrand, and every procedure that uses integral must
be aware of this. In effect, we have created two classes of procedures:
ordinary procedures and procedures that take delayed arguments. In
general, creating separate classes of procedures forces us to create sep-
arate classes of higher-order procedures as well.72
  72 is is a small reflection, in Lisp, of the difficulties that conventional strongly typed

languages such as Pascal have in coping with higher-order procedures. In such lan-
guages, the programmer must specify the data types of the arguments and the result
of each procedure: number, logical value, sequence, and so on. Consequently, we could
not express an abstraction such as “map a given procedure proc over all the elements in
a sequence” by a single higher-order procedure such as stream-map. Rather, we would
need a different mapping procedure for each different combination of argument and
result data types that might be specified for a proc. Maintaining a practical notion of
“data type” in the presence of higher-order procedures raises many difficult issues. One
way of dealing with this problem is illustrated by the language ML (Gordon et al. 1979),

     One way to avoid the need for two different classes of procedures is
to make all procedures take delayed arguments. We could adopt a model
of evaluation in which all arguments to procedures are automatically
delayed and arguments are forced only when they are actually needed
(for example, when they are required by a primitive operation). is
would transform our language to use normal-order evaluation, which
we first described when we introduced the substitution model for evalu-
ation in Section 1.1.5. Converting to normal-order evaluation provides a
uniform and elegant way to simplify the use of delayed evaluation, and
this would be a natural strategy to adopt if we were concerned only
with stream processing. In Section 4.2, aer we have studied the eval-
uator, we will see how to transform our language in just this way. Un-
fortunately, including delays in procedure calls wreaks havoc with our
ability to design programs that depend on the order of events, such as
programs that use assignment, mutate data, or perform input or output.
Even the single delay in cons-stream can cause great confusion, as
illustrated by Exercise 3.51 and Exercise 3.52. As far as anyone knows,
mutability and delayed evaluation do not mix well in programming lan-
guages, and devising ways to deal with both of these at once is an active
area of research.

whose “polymorphic data types” include templates for higher-order transformations
between data types. Moreover, data types for most procedures in ML are never explic-
itly declared by the programmer. Instead, ML includes a type-inferencing mechanism
that uses information in the environment to deduce the data types for newly defined

3.5.5 Modularity of Functional Programs
      and Modularity of Objects
As we saw in Section 3.1.2, one of the major benefits of introducing
assignment is that we can increase the modularity of our systems by
encapsulating, or “hiding,” parts of the state of a large system within
local variables. Stream models can provide an equivalent modularity
without the use of assignment. As an illustration, we can reimplement
the Monte Carlo estimation of π , which we examined in Section 3.1.2,
from a stream-processing point of view.
    e key modularity issue was that we wished to hide the internal
state of a random-number generator from programs that used random
numbers. We began with a procedure rand-update, whose successive
values furnished our supply of random numbers, and used this to pro-
duce a random-number generator:
(define rand
  (let ((x random-init))
    (lambda ()
      (set! x (rand-update x))

In the stream formulation there is no random-number generator per se,
just a stream of random numbers produced by successive calls to rand-

(define random-numbers
   (stream-map rand-update random-numbers)))

We use this to construct the stream of outcomes of the Cesàro experi-
ment performed on consecutive pairs in the random-numbers stream:

(define cesaro-stream
   (lambda (r1 r2) (= (gcd r1 r2) 1))
(define (map-successive-pairs f s)
   (f (stream-car s) (stream-car (stream-cdr s)))
   (map-successive-pairs f (stream-cdr (stream-cdr s)))))

e cesaro-stream is now fed to a monte-carlo procedure, which pro-
duces a stream of estimates of probabilities. e results are then con-
verted into a stream of estimates of π . is version of the program
doesn’t need a parameter telling how many trials to perform. Beer esti-
mates of π (from performing more experiments) are obtained by looking
farther into the pi stream:
(define (monte-carlo experiment-stream passed failed)
  (define (next passed failed)
     (/ passed (+ passed failed))
      (stream-cdr experiment-stream) passed failed)))
  (if (stream-car experiment-stream)
      (next (+ passed 1) failed)
      (next passed (+ failed 1))))
(define pi
   (lambda (p) (sqrt (/ 6 p)))
   (monte-carlo cesaro-stream 0 0)))

ere is considerable modularity in this approach, because we still can
formulate a general monte-carlo procedure that can deal with arbitrary
experiments. Yet there is no assignment or local state.

      Exercise 3.81: Exercise 3.6 discussed generalizing the random-
      number generator to allow one to reset the random-number
      sequence so as to produce repeatable sequences of “ran-
      dom” numbers. Produce a stream formulation of this same
      generator that operates on an input stream of requests to
      generate a new random number or to reset the sequence
      to a specified value and that produces the desired stream of
      random numbers. Don’t use assignment in your solution.

      Exercise 3.82: Redo Exercise 3.5 on Monte Carlo integra-
      tion in terms of streams. e stream version of estimate-
      integral will not have an argument telling how many tri-
      als to perform. Instead, it will produce a stream of estimates
      based on successively more trials.

A functional-programming view of time
Let us now return to the issues of objects and state that were raised at
the beginning of this chapter and examine them in a new light. We in-
troduced assignment and mutable objects to provide a mechanism for
modular construction of programs that model systems with state. We
constructed computational objects with local state variables and used
assignment to modify these variables. We modeled the temporal behav-
ior of the objects in the world by the temporal behavior of the corre-
sponding computational objects.
    Now we have seen that streams provide an alternative way to model
objects with local state. We can model a changing quantity, such as the
local state of some object, using a stream that represents the time his-
tory of successive states. In essence, we represent time explicitly, using
streams, so that we decouple time in our simulated world from the se-

quence of events that take place during evaluation. Indeed, because of
the presence of delay there may be lile relation between simulated
time in the model and the order of events during the evaluation.
    In order to contrast these two approaches to modeling, let us recon-
sider the implementation of a “withdrawal processor” that monitors the
balance in a bank account. In Section 3.1.3 we implemented a simplified
version of such a processor:
(define (make-simplified-withdraw balance)
  (lambda (amount)
    (set! balance (- balance amount))

Calls to make-simplified-withdraw produce computational objects,
each with a local state variable balance that is decremented by suc-
cessive calls to the object. e object takes an amount as an argument
and returns the new balance. We can imagine the user of a bank ac-
count typing a sequence of inputs to such an object and observing the
sequence of returned values shown on a display screen.
    Alternatively, we can model a withdrawal processor as a procedure
that takes as input a balance and a stream of amounts to withdraw and
produces the stream of successive balances in the account:
(define (stream-withdraw balance amount-stream)
   (stream-withdraw (- balance (stream-car amount-stream))
                      (stream-cdr amount-stream))))

stream-withdraw implements a well-defined mathematical function whose
output is fully determined by its input. Suppose, however, that the in-
put amount-stream is the stream of successive values typed by the user
and that the resulting stream of balances is displayed. en, from the

perspective of the user who is typing values and watching results, the
stream process has the same behavior as the object created by make-
simplified-withdraw. However, with the stream version, there is no
assignment, no local state variable, and consequently none of the theo-
retical difficulties that we encountered in Section 3.1.3. Yet the system
has state!
    is is really remarkable. Even though stream-withdraw implements
a well-defined mathematical function whose behavior does not change,
the user’s perception here is one of interacting with a system that has a
changing state. One way to resolve this paradox is to realize that it is the
user’s temporal existence that imposes state on the system. If the user
could step back from the interaction and think in terms of streams of
balances rather than individual transactions, the system would appear
    From the point of view of one part of a complex process, the other
parts appear to change with time. ey have hidden time-varying lo-
cal state. If we wish to write programs that model this kind of natural
decomposition in our world (as we see it from our viewpoint as a part
of that world) with structures in our computer, we make computational
objects that are not functional—they must change with time. We model
state with local state variables, and we model the changes of state with
assignments to those variables. By doing this we make the time of ex-
ecution of a computation model time in the world that we are part of,
and thus we get “objects” in our computer.
    Modeling with objects is powerful and intuitive, largely because this
matches the perception of interacting with a world of which we are
  73 Similarly  in physics, when we observe a moving particle, we say that the position
(state) of the particle is changing. However, from the perspective of the particle’s world
line in space-time there is no change involved.

part. However, as we’ve seen repeatedly throughout this chapter, these
models raise thorny problems of constraining the order of events and
of synchronizing multiple processes. e possibility of avoiding these
problems has stimulated the development of functional programming
languages, which do not include any provision for assignment or mu-
table data. In such a language, all procedures implement well-defined
mathematical functions of their arguments, whose behavior does not
change. e functional approach is extremely aractive for dealing with
concurrent systems.74
    On the other hand, if we look closely, we can see time-related prob-
lems creeping into functional models as well. One particularly trou-
blesome area arises when we wish to design interactive systems, es-
pecially ones that model interactions between independent entities. For
instance, consider once more the implementation a banking system that
permits joint bank accounts. In a conventional system using assignment
and objects, we would model the fact that Peter and Paul share an ac-
count by having both Peter and Paul send their transaction requests
to the same bank-account object, as we saw in Section 3.1.3. From the
stream point of view, where there are no “objects” per se, we have al-
ready indicated that a bank account can be modeled as a process that
operates on a stream of transaction requests to produce a stream of
responses. Accordingly, we could model the fact that Peter and Paul
have a joint bank account by merging Peter’s stream of transaction re-
quests with Paul’s stream of requests and feeding the result to the bank-
account stream process, as shown in Figure 3.38.
   74 John Backus, the inventor of Fortran, gave high visibility to functional program-

ming when he was awarded the  Turing award in 1978. His acceptance speech
(Backus 1978) strongly advocated the functional approach. A good overview of func-
tional programming is given in Henderson 1980 and in Darlington et al. 1982.

           Peter's requests                                bank
           Paul's requests                                 account

       Figure 3.38: A joint bank account, modeled by merging
       two streams of transaction requests.

    e trouble with this formulation is in the notion of merge. It will
not do to merge the two streams by simply taking alternately one re-
quest from Peter and one request from Paul. Suppose Paul accesses the
account only very rarely. We could hardly force Peter to wait for Paul to
access the account before he could issue a second transaction. However
such a merge is implemented, it must interleave the two transaction
streams in some way that is constrained by “real time” as perceived
by Peter and Paul, in the sense that, if Peter and Paul meet, they can
agree that certain transactions were processed before the meeting, and
other transactions were processed aer the meeting.75 is is precisely
the same constraint that we had to deal with in Section 3.4.1, where
we found the need to introduce explicit synchronization to ensure a
“correct” order of events in concurrent processing of objects with state.
us, in an aempt to support the functional style, the need to merge
inputs from different agents reintroduces the same problems that the
functional style was meant to eliminate.
  75 Observe that, for any two streams, there is in general more than one acceptable or-

der of interleaving. us, technically, “merge” is a relation rather than a function—the
answer is not a deterministic function of the inputs. We already mentioned (Footnote
39) that nondeterminism is essential when dealing with concurrency. e merge rela-
tion illustrates the same essential nondeterminism, from the functional perspective. In
Section 4.3, we will look at nondeterminism from yet another point of view.

    We began this chapter with the goal of building computational mod-
els whose structure matches our perception of the real world we are
trying to model. We can model the world as a collection of separate,
time-bound, interacting objects with state, or we can model the world
as a single, timeless, stateless unity. Each view has powerful advantages,
but neither view alone is completely satisfactory. A grand unification
has yet to emerge.76

  76 e  object model approximates the world by dividing it into separate pieces. e
functional model does not modularize along object boundaries. e object model is
useful when the unshared state of the “objects” is much larger than the state that they
share. An example of a place where the object viewpoint fails is quantum mechanics,
where thinking of things as individual particles leads to paradoxes and confusions. Uni-
fying the object view with the functional view may have lile to do with programming,
but rather with fundamental epistemological issues.

Metalinguistic Abstraction

     . . . It’s in words that the magic is—Abracadabra, Open Sesame,
     and the rest—but the magic words in one story aren’t magi-
     cal in the next. e real magic is to understand which words
     work, and when, and for what; the trick is to learn the trick.
     . . . And those words are made from the leers of our alpha-
     bet: a couple-dozen squiggles we can draw with the pen.
     is is the key! And the treasure, too, if we can only get
     our hands on it! It’s as if—as if the key to the treasure is the
     —John Barth, Chimera

I      , we have seen that expert program-
  mers control the complexity of their designs with the same general
techniques used by designers of all complex systems. ey combine
primitive elements to form compound objects, they abstract compound

objects to form higher-level building blocks, and they preserve modu-
larity by adopting appropriate large-scale views of system structure. In
illustrating these techniques, we have used Lisp as a language for de-
scribing processes and for constructing computational data objects and
processes to model complex phenomena in the real world. However, as
we confront increasingly complex problems, we will find that Lisp, or
indeed any fixed programming language, is not sufficient for our needs.
We must constantly turn to new languages in order to express our ideas
more effectively. Establishing new languages is a powerful strategy for
controlling complexity in engineering design; we can oen enhance our
ability to deal with a complex problem by adopting a new language that
enables us to describe (and hence to think about) the problem in a dif-
ferent way, using primitives, means of combination, and means of ab-
straction that are particularly well suited to the problem at hand.1
     Programming is endowed with a multitude of languages. ere are
   1 e   same idea is pervasive throughout all of engineering. For example, electri-
cal engineers use many different languages for describing circuits. Two of these are
the language of electrical networks and the language of electrical systems. e network
language emphasizes the physical modeling of devices in terms of discrete electrical el-
ements. e primitive objects of the network language are primitive electrical compo-
nents such as resistors, capacitors, inductors, and transistors, which are characterized
in terms of physical variables called voltage and current. When describing circuits in
the network language, the engineer is concerned with the physical characteristics of a
design. In contrast, the primitive objects of the system language are signal-processing
modules such as filters and amplifiers. Only the functional behavior of the modules is
relevant, and signals are manipulated without concern for their physical realization as
voltages and currents. e system language is erected on the network language, in the
sense that the elements of signal-processing systems are constructed from electrical
networks. Here, however, the concerns are with the large-scale organization of elec-
trical devices to solve a given application problem; the physical feasibility of the parts
is assumed. is layered collection of languages is another example of the stratified
design technique illustrated by the picture language of Section 2.2.4.

physical languages, such as the machine languages for particular com-
puters. ese languages are concerned with the representation of data
and control in terms of individual bits of storage and primitive machine
instructions. e machine-language programmer is concerned with us-
ing the given hardware to erect systems and utilities for the efficient im-
plementation of resource-limited computations. High-level languages,
erected on a machine-language substrate, hide concerns about the rep-
resentation of data as collections of bits and the representation of pro-
grams as sequences of primitive instructions. ese languages have means
of combination and abstraction, such as procedure definition, that are
appropriate to the larger-scale organization of systems.
    Metalinguistic abstraction—establishing new languages—plays an im-
portant role in all branches of engineering design. It is particularly im-
portant to computer programming, because in programming not only
can we formulate new languages but we can also implement these lan-
guages by constructing evaluators. An evaluator (or interpreter ) for a
programming language is a procedure that, when applied to an expres-
sion of the language, performs the actions required to evaluate that ex-
    It is no exaggeration to regard this as the most fundamental idea in

      e evaluator, which determines the meaning of expres-
      sions in a programming language, is just another program.

To appreciate this point is to change our images of ourselves as pro-
grammers. We come to see ourselves as designers of languages, rather
than only users of languages designed by others.
    In fact, we can regard almost any program as the evaluator for some
language. For instance, the polynomial manipulation system of Section

2.5.3 embodies the rules of polynomial arithmetic and implements them
in terms of operations on list-structured data. If we augment this system
with procedures to read and print polynomial expressions, we have the
core of a special-purpose language for dealing with problems in sym-
bolic mathematics. e digital-logic simulator of Section 3.3.4 and the
constraint propagator of Section 3.3.5 are legitimate languages in their
own right, each with its own primitives, means of combination, and
means of abstraction. Seen from this perspective, the technology for
coping with large-scale computer systems merges with the technology
for building new computer languages, and computer science itself be-
comes no more (and no less) than the discipline of constructing appro-
priate descriptive languages.
     We now embark on a tour of the technology by which languages are
established in terms of other languages. In this chapter we shall use Lisp
as a base, implementing evaluators as Lisp procedures. Lisp is particu-
larly well suited to this task, because of its ability to represent and ma-
nipulate symbolic expressions. We will take the first step in understand-
ing how languages are implemented by building an evaluator for Lisp
itself. e language implemented by our evaluator will be a subset of the
Scheme dialect of Lisp that we use in this book. Although the evaluator
described in this chapter is wrien for a particular dialect of Lisp, it con-
tains the essential structure of an evaluator for any expression-oriented
language designed for writing programs for a sequential machine. (In
fact, most language processors contain, deep within them, a lile “Lisp”
evaluator.) e evaluator has been simplified for the purposes of illus-
tration and discussion, and some features have been le out that would
be important to include in a production-quality Lisp system. Neverthe-
less, this simple evaluator is adequate to execute most of the programs

in this book.2
    An important advantage of making the evaluator accessible as a
Lisp program is that we can implement alternative evaluation rules by
describing these as modifications to the evaluator program. One place
where we can use this power to good effect is to gain extra control over
the ways in which computational models embody the notion of time,
which was so central to the discussion in Chapter 3. ere, we mitigated
some of the complexities of state and assignment by using streams to
decouple the representation of time in the world from time in the com-
puter. Our stream programs, however, were sometimes cumbersome,
because they were constrained by the applicative-order evaluation of
Scheme. In Section 4.2, we’ll change the underlying language to provide
for a more elegant approach, by modifying the evaluator to provide for
normal-order evaluation.
    Section 4.3 implements a more ambitious linguistic change, whereby
expressions have many values, rather than just a single value. In this
language of nondeterministic computing, it is natural to express processes
that generate all possible values for expressions and then search for
those values that satisfy certain constraints. In terms of models of com-
putation and time, this is like having time branch into a set of “possible
futures” and then searching for appropriate time lines. With our nonde-
terministic evaluator, keeping track of multiple values and performing
searches are handled automatically by the underlying mechanism of the
    In Section 4.4 we implement a logic-programming language in which
    2 e most important features that our evaluator leaves out are mechanisms for han-

dling errors and supporting debugging. For a more extensive discussion of evaluators,
see Friedman et al. 1992, which gives an exposition of programming languages that
proceeds via a sequence of evaluators wrien in Scheme.

knowledge is expressed in terms of relations, rather than in terms of
computations with inputs and outputs. Even though this makes the lan-
guage drastically different from Lisp, or indeed from any conventional
language, we will see that the logic-programming evaluator shares the
essential structure of the Lisp evaluator.

4.1 The Metacircular Evaluator
Our evaluator for Lisp will be implemented as a Lisp program. It may
seem circular to think about evaluating Lisp programs using an evalua-
tor that is itself implemented in Lisp. However, evaluation is a process,
so it is appropriate to describe the evaluation process using Lisp, which,
aer all, is our tool for describing processes.3 An evaluator that is writ-
ten in the same language that it evaluates is said to be metacircular.
     e metacircular evaluator is essentially a Scheme formulation of
the environment model of evaluation described in Section 3.2. Recall
that the model has two basic parts:

    1. To evaluate a combination (a compound expression other than
       a special form), evaluate the subexpressions and then apply the
       value of the operator subexpression to the values of the operand

    2. To apply a compound procedure to a set of arguments, evaluate
       the body of the procedure in a new environment. To construct
   3 Even so, there will remain important aspects of the evaluation process that are not

elucidated by our evaluator. e most important of these are the detailed mechanisms
by which procedures call other procedures and return values to their callers. We will
address these issues in Chapter 5, where we take a closer look at the evaluation process
by implementing the evaluator as a simple register machine.

       this environment, extend the environment part of the procedure
       object by a frame in which the formal parameters of the procedure
       are bound to the arguments to which the procedure is applied.

ese two rules describe the essence of the evaluation process, a basic
cycle in which expressions to be evaluated in environments are reduced
to procedures to be applied to arguments, which in turn are reduced to
new expressions to be evaluated in new environments, and so on, un-
til we get down to symbols, whose values are looked up in the envi-
ronment, and to primitive procedures, which are applied directly (see
Figure 4.1).4 is evaluation cycle will be embodied by the interplay
between the two critical procedures in the evaluator, eval and apply,
which are described in Section 4.1.1 (see Figure 4.1).
    e implementation of the evaluator will depend upon procedures
    4 If we grant ourselves the ability to apply primitives, then what remains for us to

implement in the evaluator? e job of the evaluator is not to specify the primitives of
the language, but rather to provide the connective tissue—the means of combination
and the means of abstraction—that binds a collection of primitives to form a language.
   • e evaluator enables us to deal with nested expressions. For example, although
simply applying primitives would suffice for evaluating the expression (+ 1 6), it is not
adequate for handling (+ 1 (* 2 3)). As far as the primitive procedure + is concerned,
its arguments must be numbers, and it would choke if we passed it the expression (*
2 3) as an argument. One important role of the evaluator is to choreograph procedure
composition so that (* 2 3) is reduced to 6 before being passed as an argument to +.
   • e evaluator allows us to use variables. For example, the primitive procedure for
addition has no way to deal with expressions such as (+ x 1). We need an evaluator to
keep track of variables and obtain their values before invoking the primitive procedures.
   • e evaluator allows us to define compound procedures. is involves keeping
track of procedure definitions, knowing how to use these definitions in evaluating ex-
pressions, and providing a mechanism that enables procedures to accept arguments.
   • e evaluator provides the special forms, which must be evaluated differently from
procedure calls.

   Procedure,                                             Expression,
                           Eval         Apply
   Arguments                                              Environment

      Figure 4.1: e eval-apply cycle exposes the essence of a
      computer language.

that define the syntax of the expressions to be evaluated. We will use
data abstraction to make the evaluator independent of the representa-
tion of the language. For example, rather than commiing to a choice
that an assignment is to be represented by a list beginning with the
symbol set! we use an abstract predicate assignment? to test for an
assignment, and we use abstract selectors assignment-variable and
assignment-value to access the parts of an assignment. Implementa-
tion of expressions will be described in detail in Section 4.1.2. ere are
also operations, described in Section 4.1.3, that specify the represen-
tation of procedures and environments. For example, make-procedure
constructs compound procedures, lookup-variable-value accesses the
values of variables, and apply-primitive-procedure applies a primi-
tive procedure to a given list of arguments.

4.1.1 The Core of the Evaluator
e evaluation process can be described as the interplay between two
procedures: eval and apply.

eval  takes as arguments an expression and an environment. It classi-
fies the expression and directs its evaluation. eval is structured as a case
analysis of the syntactic type of the expression to be evaluated. In or-
der to keep the procedure general, we express the determination of the
type of an expression abstractly, making no commitment to any partic-
ular representation for the various types of expressions. Each type of
expression has a predicate that tests for it and an abstract means for
selecting its parts. is abstract syntax makes it easy to see how we can
change the syntax of the language by using the same evaluator, but with
a different collection of syntax procedures.

Primitive expressions
    • For self-evaluating expressions, such as numbers, eval returns
      the expression itself.
    • eval must look up variables in the environment to find their val-
Special forms
    • For quoted expressions, eval returns the expression that was quoted.
    • An assignment to (or a definition o) a variable must recursively
      call eval to compute the new value to be associated with the vari-
      able. e environment must be modified to change (or create) the
      binding of the variable.

    • An if expression requires special processing of its parts, so as to
      evaluate the consequent if the predicate is true, and otherwise to
      evaluate the alternative.

    • A lambda expression must be transformed into an applicable pro-
      cedure by packaging together the parameters and body specified
      by the lambda expression with the environment of the evaluation.

    • A begin expression requires evaluating its sequence of expres-
      sions in the order in which they appear.

    • A case analysis (cond) is transformed into a nest of if expressions
      and then evaluated.


    • For a procedure application, eval must recursively evaluate the
      operator part and the operands of the combination. e resulting
      procedure and arguments are passed to apply, which handles the
      actual procedure application.

Here is the definition of eval:
(define (eval exp env)
  (cond ((self-evaluating? exp) exp)
        ((variable? exp) (lookup-variable-value exp env))
        ((quoted? exp) (text-of-quotation exp))
        ((assignment? exp) (eval-assignment exp env))
        ((definition? exp) (eval-definition exp env))
        ((if? exp) (eval-if exp env))
        ((lambda? exp) (make-procedure (lambda-parameters exp)
                                           (lambda-body exp)

        ((begin? exp)
          (eval-sequence (begin-actions exp) env))
        ((cond? exp) (eval (cond->if exp) env))
        ((application? exp)
          (apply (eval (operator exp) env)
                 (list-of-values (operands exp) env)))
          (error "Unknown expression type: EVAL" exp))))

For clarity, eval has been implemented as a case analysis using cond.
e disadvantage of this is that our procedure handles only a few distin-
guishable types of expressions, and no new ones can be defined without
editing the definition of eval. In most Lisp implementations, dispatch-
ing on the type of an expression is done in a data-directed style. is
allows a user to add new types of expressions that eval can distinguish,
without modifying the definition of eval itself. (See Exercise 4.3.)

apply  takes two arguments, a procedure and a list of arguments to
which the procedure should be applied. apply classifies procedures into
two kinds: It calls apply-primitive-procedure to apply primitives; it
applies compound procedures by sequentially evaluating the expres-
sions that make up the body of the procedure. e environment for
the evaluation of the body of a compound procedure is constructed by
extending the base environment carried by the procedure to include a
frame that binds the parameters of the procedure to the arguments to
which the procedure is to be applied. Here is the definition of apply:
(define (apply procedure arguments)
  (cond ((primitive-procedure? procedure)
          (apply-primitive-procedure procedure arguments))

          ((compound-procedure? procedure)
              (procedure-body procedure)
                 (procedure-parameters procedure)
                 (procedure-environment procedure))))
             "Unknown procedure type: APPLY" procedure))))

Procedure arguments
When eval processes a procedure application, it uses list-of-values
to produce the list of arguments to which the procedure is to be applied.
list-of-values takes as an argument the operands of the combina-
tion. It evaluates each operand and returns a list of the corresponding
(define (list-of-values exps env)
  (if (no-operands? exps)
       (cons (eval (first-operand exps) env)
               (list-of-values (rest-operands exps) env))))

   5 We could have simplified the application? clause in eval by using map (and stip-

ulating that operands returns a list) rather than writing an explicit list-of-values
procedure. We chose not to use map here to emphasize the fact that the evaluator can
be implemented without any use of higher-order procedures (and thus could be writ-
ten in a language that doesn’t have higher-order procedures), even though the language
that it supports will include higher-order procedures.

eval-if evaluates the predicate part of an if expression in the given
environment. If the result is true, eval-if evaluates the consequent,
otherwise it evaluates the alternative:
(define (eval-if exp env)
  (if (true? (eval (if-predicate exp) env))
       (eval (if-consequent exp) env)
       (eval (if-alternative exp) env)))

e use of true? in eval-if highlights the issue of the connection be-
tween an implemented language and an implementation language. e
if-predicate is evaluated in the language being implemented and thus
yields a value in that language. e interpreter predicate true? trans-
lates that value into a value that can be tested by the if in the imple-
mentation language: e metacircular representation of truth might not
be the same as that of the underlying Scheme.6

eval-sequence is used by apply to evaluate the sequence of expressions
in a procedure body and by eval to evaluate the sequence of expressions
in a begin expression. It takes as arguments a sequence of expressions
and an environment, and evaluates the expressions in the order in which
they occur. e value returned is the value of the final expression.
(define (eval-sequence exps env)
  (cond ((last-exp? exps)
           (eval (first-exp exps) env))

   6 In this case, the language being implemented and the implementation language are

the same. Contemplation of the meaning of true? here yields expansion of conscious-
ness without the abuse of substance.

           (eval (first-exp exps) env)
           (eval-sequence (rest-exps exps) env))))

Assignments and definitions
e following procedure handles assignments to variables. It calls eval
to find the value to be assigned and transmits the variable and the re-
sulting value to set-variable-value! to be installed in the designated
(define (eval-assignment exp env)
  (set-variable-value! (assignment-variable exp)
                              (eval (assignment-value exp) env)

Definitions of variables are handled in a similar manner.7
(define (eval-definition exp env)
  (define-variable! (definition-variable exp)
                          (eval (definition-value exp) env)

We have chosen here to return the symbol ok as the value of an assign-
ment or a definition.8

       Exercise 4.1: Notice that we cannot tell whether the metacir-
       cular evaluator evaluates operands from le to right or from
   7 is implementation of define ignores a subtle issue in the handling of internal
definitions, although it works correctly in most cases. We will see what the problem is
and how to solve it in Section 4.1.6.
   8 As we said when we introduced define and set!, these values are implementation-

dependent in Scheme—that is, the implementor can choose what value to return.

      right to le. Its evaluation order is inherited from the un-
      derlying Lisp: If the arguments to cons in list-of-values
      are evaluated from le to right, then list-of-values will
      evaluate operands from le to right; and if the arguments to
      cons are evaluated from right to le, then list-of-values
      will evaluate operands from right to le.
      Write a version of list-of-values that evaluates operands
      from le to right regardless of the order of evaluation in the
      underlying Lisp. Also write a version of list-of-values
      that evaluates operands from right to le.

4.1.2 Representing Expressions
e evaluator is reminiscent of the symbolic differentiation program
discussed in Section 2.3.2. Both programs operate on symbolic expres-
sions. In both programs, the result of operating on a compound expres-
sion is determined by operating recursively on the pieces of the expres-
sion and combining the results in a way that depends on the type of
the expression. In both programs we used data abstraction to decouple
the general rules of operation from the details of how expressions are
represented. In the differentiation program this meant that the same
differentiation procedure could deal with algebraic expressions in pre-
fix form, in infix form, or in some other form. For the evaluator, this
means that the syntax of the language being evaluated is determined
solely by the procedures that classify and extract pieces of expressions.
    Here is the specification of the syntax of our language:

    • e only self-evaluating items are numbers and strings:
      (define (self-evaluating? exp)
        (cond ((number? exp) true)

                  ((string? exp) true)
                  (else false)))

    • Variables are represented by symbols:
       (define (variable? exp) (symbol? exp))

    • otations have the form (quote ⟨text-of-quotation⟩):9
       (define (quoted? exp) (tagged-list? exp 'quote))
       (define (text-of-quotation exp) (cadr exp))

       quoted? is defined in terms of the procedure tagged-list?, which
       identifies lists beginning with a designated symbol:
       (define (tagged-list? exp tag)
          (if (pair? exp)
               (eq? (car exp) tag)

    • Assignments have the form (set! ⟨var⟩ ⟨value⟩):
       (define (assignment? exp) (tagged-list? exp 'set!))
       (define (assignment-variable exp) (cadr exp))
       (define (assignment-value exp) (caddr exp))

    • Definitions have the form
       (define    ⟨var⟩ ⟨value⟩)

       or the form
   9 As mentioned in Section 2.3.1, the evaluator sees a quoted expression as a list begin-

ning with quote, even if the expression is typed with the quotation mark. For example,
the expression 'a would be seen by the evaluator as (quote a). See Exercise 2.55.

  (define (⟨var⟩    ⟨parameter1 ⟩ . . . ⟨parametern ⟩)

  e laer form (standard procedure definition) is syntactic sugar
  (define   ⟨var⟩
    (lambda (⟨parameter1 ⟩    . . . ⟨parametern ⟩)

  e corresponding syntax procedures are the following:
  (define (definition? exp) (tagged-list? exp 'define))
  (define (definition-variable exp)
    (if (symbol? (cadr exp))
        (cadr exp)
        (caadr exp)))
  (define (definition-value exp)
    (if (symbol? (cadr exp))
        (caddr exp)
        (make-lambda (cdadr exp)            ; formal parameters
                         (cddr exp))))      ; body

• lambda expressions are lists that begin with the symbol lambda:
  (define (lambda? exp) (tagged-list? exp 'lambda))
  (define (lambda-parameters exp) (cadr exp))
  (define (lambda-body exp) (cddr exp))

  We also provide a constructor for lambda expressions, which is
  used by definition-value, above:
  (define (make-lambda parameters body)
    (cons 'lambda (cons parameters body)))

     • Conditionals begin with if and have a predicate, a consequent,
       and an (optional) alternative. If the expression has no alternative
       part, we provide false as the alternative.10
       (define (if? exp) (tagged-list? exp 'if))
       (define (if-predicate exp) (cadr exp))
       (define (if-consequent exp) (caddr exp))
       (define (if-alternative exp)
          (if (not (null? (cdddr exp)))
                (cadddr exp)

       We also provide a constructor for if expressions, to be used by
       cond->if to transform cond expressions into if expressions:

       (define (make-if predicate consequent alternative)
          (list 'if predicate consequent alternative))

     • begin packages a sequence of expressions into a single expres-
       sion. We include syntax operations on begin expressions to ex-
       tract the actual sequence from the begin expression, as well as
       selectors that return the first expression and the rest of the ex-
       pressions in the sequence.11
       (define (begin? exp) (tagged-list? exp 'begin))
       (define (begin-actions exp) (cdr exp))

  10 e value of an if expression when the predicate is false and there is no alternative

is unspecified in Scheme; we have chosen here to make it false. We will support the use
of the variables true and false in expressions to be evaluated by binding them in the
global environment. See Section 4.1.4.
   11 ese selectors for a list of expressions—and the corresponding ones for a list of

operands—are not intended as a data abstraction. ey are introduced as mnemonic
names for the basic list operations in order to make it easier to understand the explicit-
control evaluator in Section 5.4.

      (define (last-exp? seq) (null? (cdr seq)))
      (define (first-exp seq) (car seq))
      (define (rest-exps seq) (cdr seq))

      We also include a constructor sequence->exp (for use by cond-
      >if)that transforms a sequence into a single expression, using
      begin if necessary:

      (define (sequence->exp seq)
        (cond ((null? seq) seq)
              ((last-exp? seq) (first-exp seq))
              (else (make-begin seq))))
      (define (make-begin seq) (cons 'begin seq))

    • A procedure application is any compound expression that is not
      one of the above expression types. e car of the expression is
      the operator, and the cdr is the list of operands:
      (define (application? exp) (pair? exp))
      (define (operator exp) (car exp))
      (define (operands exp) (cdr exp))
      (define (no-operands? ops) (null? ops))
      (define (first-operand ops) (car ops))
      (define (rest-operands ops) (cdr ops))

Derived expressions
Some special forms in our language can be defined in terms of expres-
sions involving other special forms, rather than being implemented di-
rectly. One example is cond, which can be implemented as a nest of if
expressions. For example, we can reduce the problem of evaluating the

(cond ((> x 0) x)
           ((= x 0) (display 'zero) 0)
           (else (- x)))

to the problem of evaluating the following expression involving if and
begin expressions:
(if (> x 0)
       (if (= x 0)
            (begin (display 'zero) 0)
            (- x)))

Implementing the evaluation of cond in this way simplifies the evalua-
tor because it reduces the number of special forms for which the evalu-
ation process must be explicitly specified.
    We include syntax procedures that extract the parts of a cond ex-
pression, and a procedure cond->if that transforms cond expressions
into if expressions. A case analysis begins with cond and has a list of
predicate-action clauses. A clause is an else clause if its predicate is the
symbol else.12
(define (cond? exp) (tagged-list? exp 'cond))
(define (cond-clauses exp) (cdr exp))
(define (cond-else-clause? clause)
  (eq? (cond-predicate clause) 'else))
(define (cond-predicate clause) (car clause))
(define (cond-actions clause) (cdr clause))
(define (cond->if exp) (expand-clauses (cond-clauses exp)))
(define (expand-clauses clauses)
  (if (null? clauses)
           'false                                ; no else clause
  12 e    value of a cond expression when all the predicates are false and there is no
else   clause is unspecified in Scheme; we have chosen here to make it false.

       (let ((first (car clauses))
               (rest (cdr clauses)))
          (if (cond-else-clause? first)
               (if (null? rest)
                     (sequence->exp (cond-actions first))
                     (error "ELSE clause isn't last: COND->IF"
               (make-if (cond-predicate first)
                           (sequence->exp (cond-actions first))
                           (expand-clauses rest))))))

Expressions (such as cond) that we choose to implement as syntactic
transformations are called derived expressions. let expressions are also
derived expressions (see Exercise 4.6).13

       Exercise 4.2: Louis Reasoner plans to reorder the cond clauses
       in eval so that the clause for procedure applications ap-
       pears before the clause for assignments. He argues that this
       will make the interpreter more efficient: Since programs
       usually contain more applications than assignments, def-
       initions, and so on, his modified eval will usually check
       fewer clauses than the original eval before identifying the
       type of an expression.

          a. What is wrong with Louis’s plan? (Hint: What will
   13 Practical Lisp systems provide a mechanism that allows a user to add new de-

rived expressions and specify their implementation as syntactic transformations with-
out modifying the evaluator. Such a user-defined transformation is called a macro. Al-
though it is easy to add an elementary mechanism for defining macros, the result-
ing language has subtle name-conflict problems. ere has been much research on
mechanisms for macro definition that do not cause these difficulties. See, for example,
Kohlbecker 1986, Clinger and Rees 1991, and Hanson 1991.

     Louis’s evaluator do with the expression (define x
  b. Louis is upset that his plan didn’t work. He is will-
     ing to go to any lengths to make his evaluator recog-
     nize procedure applications before it checks for most
     other kinds of expressions. Help him by changing the
     syntax of the evaluated language so that procedure
     applications start with call. For example, instead of
     (factorial 3) we will now have to write (call factorial
     3) and instead of (+ 1 2) we will have to write (call
     + 1 2).

Exercise 4.3: Rewrite eval so that the dispatch is done
in data-directed style. Compare this with the data-directed
differentiation procedure of Exercise 2.73. (You may use the
car of a compound expression as the type of the expres-
sion, as is appropriate for the syntax implemented in this

Exercise 4.4: Recall the definitions of the special forms and
and or from Chapter 1:

   • and: e expressions are evaluated from le to right.
     If any expression evaluates to false, false is returned;
     any remaining expressions are not evaluated. If all the
     expressions evaluate to true values, the value of the
     last expression is returned. If there are no expressions
     then true is returned.
   • or: e expressions are evaluated from le to right.
     If any expression evaluates to a true value, that value

     is returned; any remaining expressions are not evalu-
     ated. If all expressions evaluate to false, or if there are
     no expressions, then false is returned.

Install and and or as new special forms for the evaluator by
defining appropriate syntax procedures and evaluation pro-
cedures eval-and and eval-or. Alternatively, show how to
implement and and or as derived expressions.
Exercise 4.5: Scheme allows an additional syntax for cond
clauses, (⟨test⟩ => ⟨recipient⟩). If ⟨test ⟩ evaluates to a
true value, then ⟨recipient ⟩ is evaluated. Its value must be a
procedure of one argument; this procedure is then invoked
on the value of the ⟨test ⟩, and the result is returned as the
value of the cond expression. For example
(cond ((assoc 'b '((a 1) (b 2))) => cadr)
       (else false))

returns 2. Modify the handling of cond so that it supports
this extended syntax.
Exercise 4.6: let expressions are derived expressions, be-
(let ((⟨var1 ⟩   ⟨exp1 ⟩) . . . (⟨varn ⟩ ⟨expn ⟩))

is equivalent to
((lambda (⟨var1 ⟩   . . . ⟨varn ⟩)
 ⟨exp1 ⟩
 ⟨expn ⟩)

Implement a syntactic transformation let->combination
that reduces evaluating let expressions to evaluating com-
binations of the type shown above, and add the appropriate
clause to eval to handle let expressions.

Exercise 4.7: let* is similar to let, except that the bind-
ings of the let* variables are performed sequentially from
le to right, and each binding is made in an environment in
which all of the preceding bindings are visible. For example
(let* ((x 3)    (y (+ x 2))    (z (+ x y 5)))
  (* x z))

returns 39. Explain how a let* expression can be rewrien
as a set of nested let expressions, and write a procedure
let*->nested-lets that performs this transformation. If
we have already implemented let (Exercise 4.6) and we
want to extend the evaluator to handle let*, is it sufficient
to add a clause to eval whose action is
(eval (let*->nested-lets exp) env)

or must we explicitly expand let* in terms of non-derived

Exercise 4.8: “Named let” is a variant of let that has the
(let   ⟨var⟩ ⟨bindings⟩ ⟨body⟩)

e ⟨bindings ⟩ and ⟨body ⟩ are just as in ordinary let, ex-
cept that ⟨var ⟩ is bound within ⟨body ⟩ to a procedure whose
body is ⟨body ⟩ and whose parameters are the variables in

the ⟨bindings ⟩. us, one can repeatedly execute the ⟨body ⟩
by invoking the procedure named ⟨var ⟩. For example, the
iterative Fibonacci procedure (Section 1.2.2) can be rewrit-
ten using named let as follows:

(define (fib n)
  (let fib-iter ((a 1)
                    (b 0)
                    (count n))
    (if (= count 0)
         (fib-iter (+ a b) a (- count 1)))))

Modify let->combination of Exercise 4.6 to also support
named let.

Exercise 4.9: Many languages support a variety of iteration
constructs, such as do, for, while, and until. In Scheme,
iterative processes can be expressed in terms of ordinary
procedure calls, so special iteration constructs provide no
essential gain in computational power. On the other hand,
such constructs are oen convenient. Design some itera-
tion constructs, give examples of their use, and show how
to implement them as derived expressions.

Exercise 4.10: By using data abstraction, we were able to
write an eval procedure that is independent of the particu-
lar syntax of the language to be evaluated. To illustrate this,
design and implement a new syntax for Scheme by modify-
ing the procedures in this section, without changing eval
or apply.

4.1.3 Evaluator Data Structures
In addition to defining the external syntax of expressions, the evaluator
implementation must also define the data structures that the evaluator
manipulates internally, as part of the execution of a program, such as the
representation of procedures and environments and the representation
of true and false.

Testing of predicates
For conditionals, we accept anything to be true that is not the explicit
false object.

(define (true? x)    (not (eq? x false)))
(define (false? x) (eq? x false))

Representing procedures
To handle primitives, we assume that we have available the following

    • (apply-primitive-procedure ⟨proc⟩ ⟨args⟩)
      applies the given primitive procedure to the argument values in
      the list ⟨args ⟩ and returns the result of the application.

    • (primitive-procedure? ⟨proc⟩)
      tests whether ⟨proc ⟩ is a primitive procedure.

ese mechanisms for handling primitives are further described in Sec-
tion 4.1.4.
    Compound procedures are constructed from parameters, procedure
bodies, and environments using the constructor make-procedure:

(define (make-procedure parameters body env)
  (list 'procedure parameters body env))
(define (compound-procedure? p)
  (tagged-list? p 'procedure))
(define (procedure-parameters p) (cadr p))
(define (procedure-body p) (caddr p))
(define (procedure-environment p) (cadddr p))

Operations on Environments
e evaluator needs operations for manipulating environments. As ex-
plained in Section 3.2, an environment is a sequence of frames, where
each frame is a table of bindings that associate variables with their cor-
responding values. We use the following operations for manipulating

    • (lookup-variable-value ⟨var⟩ ⟨env⟩)
      returns the value that is bound to the symbol ⟨var ⟩ in the envi-
      ronment ⟨env ⟩, or signals an error if the variable is unbound.

    • (extend-environment ⟨variables⟩ ⟨values⟩ ⟨base-env⟩)
      returns a new environment, consisting of a new frame in which
      the symbols in the list ⟨variables ⟩ are bound to the corresponding
      elements in the list ⟨values ⟩, where the enclosing environment is
      the environment ⟨base-env ⟩.

    • (define-variable! ⟨var⟩ ⟨value⟩ ⟨env⟩)
      adds to the first frame in the environment ⟨env ⟩ a new binding
      that associates the variable ⟨var ⟩ with the value ⟨value ⟩.

    • (set-variable-value! ⟨var⟩ ⟨value⟩ ⟨env⟩)
         changes the binding of the variable ⟨var ⟩ in the environment ⟨env ⟩
         so that the variable is now bound to the value ⟨value ⟩, or signals
         an error if the variable is unbound.

To implement these operations we represent an environment as a list of
frames. e enclosing environment of an environment is the cdr of the
list. e empty environment is simply the empty list.
(define (enclosing-environment env) (cdr env))
(define (first-frame env) (car env))
(define the-empty-environment '())

Each frame of an environment is represented as a pair of lists: a list of
the variables bound in that frame and a list of the associated values.14
(define (make-frame variables values)
  (cons variables values))
(define (frame-variables frame) (car frame))
(define (frame-values frame) (cdr frame))
(define (add-binding-to-frame! var val frame)
  (set-car! frame (cons var (car frame)))
  (set-cdr! frame (cons val (cdr frame))))

To extend an environment by a new frame that associates variables with
values, we make a frame consisting of the list of variables and the list
of values, and we adjoin this to the environment. We signal an error if
the number of variables does not match the number of values.
  14 Frames are not really a data abstraction in the following code: set-variable-
value!  and define-variable! use set-car! to directly modify the values in a frame.
e purpose of the frame procedures is to make the environment-manipulation proce-
dures easy to read.

(define (extend-environment vars vals base-env)
  (if (= (length vars) (length vals))
      (cons (make-frame vars vals) base-env)
      (if (< (length vars) (length vals))
          (error "Too many arguments supplied" vars vals)
          (error "Too few arguments supplied" vars vals))))

To look up a variable in an environment, we scan the list of variables
in the first frame. If we find the desired variable, we return the corre-
sponding element in the list of values. If we do not find the variable
in the current frame, we search the enclosing environment, and so on.
If we reach the empty environment, we signal an “unbound variable”
(define (lookup-variable-value var env)
  (define (env-loop env)
    (define (scan vars vals)
      (cond ((null? vars)
              (env-loop (enclosing-environment env)))
             ((eq? var (car vars)) (car vals))
             (else (scan (cdr vars) (cdr vals)))))
    (if (eq? env the-empty-environment)
        (error "Unbound variable" var)
        (let ((frame (first-frame env)))
          (scan (frame-variables frame)
                 (frame-values frame)))))
  (env-loop env))

To set a variable to a new value in a specified environment, we scan for
the variable, just as in lookup-variable-value, and change the corre-
sponding value when we find it.
(define (set-variable-value! var val env)
  (define (env-loop env)

    (define (scan vars vals)
      (cond ((null? vars)
              (env-loop (enclosing-environment env)))
             ((eq? var (car vars)) (set-car! vals val))
             (else (scan (cdr vars) (cdr vals)))))
    (if (eq? env the-empty-environment)
        (error "Unbound variable: SET!" var)
        (let ((frame (first-frame env)))
           (scan (frame-variables frame)
                 (frame-values frame)))))
  (env-loop env))

To define a variable, we search the first frame for a binding for the
variable, and change the binding if it exists (just as in set-variable-
value!). If no such binding exists, we adjoin one to the first frame.

(define (define-variable! var val env)
  (let ((frame (first-frame env)))
    (define (scan vars vals)
      (cond ((null? vars)
              (add-binding-to-frame! var val frame))
             ((eq? var (car vars)) (set-car! vals val))
             (else (scan (cdr vars) (cdr vals)))))
    (scan (frame-variables frame) (frame-values frame))))

e method described here is only one of many plausible ways to rep-
resent environments. Since we used data abstraction to isolate the rest
of the evaluator from the detailed choice of representation, we could
change the environment representation if we wanted to. (See Exercise
4.11.) In a production-quality Lisp system, the speed of the evaluator’s
environment operations—especially that of variable lookup—has a ma-
jor impact on the performance of the system. e representation de-
scribed here, although conceptually simple, is not efficient and would

not ordinarily be used in a production system.15

       Exercise 4.11: Instead of representing a frame as a pair of
       lists, we can represent a frame as a list of bindings, where
       each binding is a name-value pair. Rewrite the environment
       operations to use this alternative representation.

       Exercise 4.12: e procedures set-variable-value!, define-
       variable! and lookup-variable-value can be expressed
       in terms of more abstract procedures for traversing the en-
       vironment structure. Define abstractions that capture the
       common paerns and redefine the three procedures in terms
       of these abstractions.

       Exercise 4.13: Scheme allows us to create new bindings for
       variables by means of define, but provides no way to get
       rid of bindings. Implement for the evaluator a special form
       make-unbound! that removes the binding of a given symbol
       from the environment in which the make-unbound! expres-
       sion is evaluated. is problem is not completely specified.
       For example, should we remove only the binding in the first
       frame of the environment? Complete the specification and
       justify any choices you make.

  15 e drawback of this representation (as well as the variant in Exercise 4.11) is that

the evaluator may have to search through many frames in order to find the binding for
a given variable. (Such an approach is referred to as deep binding.) One way to avoid this
inefficiency is to make use of a strategy called lexical addressing, which will be discussed
in Section 5.5.6.

4.1.4 Running the Evaluator as a Program
Given the evaluator, we have in our hands a description (expressed in
Lisp) of the process by which Lisp expressions are evaluated. One ad-
vantage of expressing the evaluator as a program is that we can run the
program. is gives us, running within Lisp, a working model of how
Lisp itself evaluates expressions. is can serve as a framework for ex-
perimenting with evaluation rules, as we shall do later in this chapter.
    Our evaluator program reduces expressions ultimately to the appli-
cation of primitive procedures. erefore, all that we need to run the
evaluator is to create a mechanism that calls on the underlying Lisp
system to model the application of primitive procedures.
    ere must be a binding for each primitive procedure name, so that
when eval evaluates the operator of an application of a primitive, it will
find an object to pass to apply. We thus set up a global environment that
associates unique objects with the names of the primitive procedures
that can appear in the expressions we will be evaluating. e global
environment also includes bindings for the symbols true and false, so
that they can be used as variables in expressions to be evaluated.
(define (setup-environment)
  (let ((initial-env
          (extend-environment (primitive-procedure-names)
    (define-variable! 'true true initial-env)
    (define-variable! 'false false initial-env)
(define the-global-environment (setup-environment))

It does not maer how we represent the primitive procedure objects,
so long as apply can identify and apply them by using the procedures

primitive-procedure? and apply-primitive-procedure. We have cho-
sen to represent a primitive procedure as a list beginning with the sym-
bol primitive and containing a procedure in the underlying Lisp that
implements that primitive.
(define (primitive-procedure? proc)
  (tagged-list? proc 'primitive))
(define (primitive-implementation proc) (cadr proc))

setup-environment    will get the primitive names and implementation
procedures from a list:16
(define primitive-procedures
  (list (list 'car car)
           (list 'cdr cdr)
           (list 'cons cons)
           (list 'null? null?)
           ⟨more primitives⟩ ))
(define (primitive-procedure-names)
  (map car primitive-procedures))
(define (primitive-procedure-objects)
  (map (lambda (proc) (list 'primitive (cadr proc)))

To apply a primitive procedure, we simply apply the implementation
procedure to the arguments, using the underlying Lisp system:17
  16 Any  procedure defined in the underlying Lisp can be used as a primitive for the
metacircular evaluator. e name of a primitive installed in the evaluator need not
be the same as the name of its implementation in the underlying Lisp; the names are
the same here because the metacircular evaluator implements Scheme itself. us, for
example, we could put (list 'first car) or (list 'square (lambda (x) (* x
x))) in the list of primitive-procedures.
  17 apply-in-underlying-scheme is the apply procedure we have used in earlier

chapters. e metacircular evaluator’s apply procedure (Section 4.1.1) models the

(define (apply-primitive-procedure proc args)
    (primitive-implementation proc) args))

For convenience in running the metacircular evaluator, we provide a
driver loop that models the read-eval-print loop of the underlying Lisp
system. It prints a prompt, reads an input expression, evaluates this ex-
pression in the global environment, and prints the result. We precede
each printed result by an output prompt so as to distinguish the value of
the expression from other output that may be printed.18
(define input-prompt         ";;; M-Eval input:")
(define output-prompt ";;; M-Eval value:")
(define (driver-loop)
  (prompt-for-input input-prompt)
  (let ((input (read)))
     (let ((output (eval input the-global-environment)))
        (announce-output output-prompt)
        (user-print output)))

working of this primitive. Having two different things called apply leads to a tech-
nical problem in running the metacircular evaluator, because defining the metacircular
evaluator’s apply will mask the definition of the primitive. One way around this is to
rename the metacircular apply to avoid conflict with the name of the primitive proce-
dure. We have assumed instead that we have saved a reference to the underlying apply
by doing
(define apply-in-underlying-scheme apply)

before defining the metacircular apply. is allows us to access the original version of
apply under a different name.
   18 e primitive procedure read waits for input from the user, and returns the next

complete expression that is typed. For example, if the user types (+ 23 x), read returns
a three-element list containing the symbol +, the number 23, and the symbol x. If the
user types 'x, read returns a two-element list containing the symbol quote and the
symbol x.

(define (prompt-for-input string)
  (newline) (newline) (display string) (newline))
(define (announce-output string)
  (newline) (display string) (newline))

We use a special printing procedure, user-print, to avoid printing the
environment part of a compound procedure, which may be a very long
list (or may even contain cycles).
(define (user-print object)
  (if (compound-procedure? object)
      (display (list 'compound-procedure
                       (procedure-parameters object)
                       (procedure-body object)
      (display object)))

Now all we need to do to run the evaluator is to initialize the global
environment and start the driver loop. Here is a sample interaction:
(define the-global-environment (setup-environment))

;;; M-Eval input:
(define (append x y)
  (if (null? x)
      (cons (car x) (append (cdr x) y))))
;;; M-Eval value:
;;; M-Eval input:
(append '(a b c) '(d e f))
;;; M-Eval value:
(a b c d e f)

      Exercise 4.14: Eva Lu Ator and Louis Reasoner are each
      experimenting with the metacircular evaluator. Eva types
      in the definition of map, and runs some test programs that
      use it. ey work fine. Louis, in contrast, has installed the
      system version of map as a primitive for the metacircular
      evaluator. When he tries it, things go terribly wrong. Ex-
      plain why Louis’s map fails even though Eva’s works.

4.1.5 Data as Programs
In thinking about a Lisp program that evaluates Lisp expressions, an
analogy might be helpful. One operational view of the meaning of a
program is that a program is a description of an abstract (perhaps in-
finitely large) machine. For example, consider the familiar program to
compute factorials:
(define (factorial n)
  (if (= n 1) 1 (* (factorial (- n 1)) n)))

We may regard this program as the description of a machine contain-
ing parts that decrement, multiply, and test for equality, together with
a two-position switch and another factorial machine. (e factorial ma-
chine is infinite because it contains another factorial machine within it.)
Figure 4.2 is a flow diagram for the factorial machine, showing how the
parts are wired together.
    In a similar way, we can regard the evaluator as a very special ma-
chine that takes as input a description of a machine. Given this input,
the evaluator configures itself to emulate the machine described. For ex-
ample, if we feed our evaluator the definition of factorial, as shown
in Figure 4.3, the evaluator will be able to compute factorials.

                      factorial          1                1

          6                              =                                 720


                          --          factorial


       Figure 4.2: e factorial program, viewed as an abstract

    From this perspective, our evaluator is seen to be a universal ma-
chine. It mimics other machines when these are described as Lisp pro-
grams.19 is is striking. Try to imagine an analogous evaluator for
  19 e  fact that the machines are described in Lisp is inessential. If we give our eval-
uator a Lisp program that behaves as an evaluator for some other language, say C,
the Lisp evaluator will emulate the C evaluator, which in turn can emulate any ma-
chine described as a C program. Similarly, writing a Lisp evaluator in C produces a C
program that can execute any Lisp program. e deep idea here is that any evaluator
can emulate any other. us, the notion of “what can in principle be computed” (ig-
noring practicalities of time and memory required) is independent of the language or
the computer, and instead reflects an underlying notion of computability. is was first
demonstrated in a clear way by Alan M. Turing (1912-1954), whose 1936 paper laid the
foundations for theoretical computer science. In the paper, Turing presented a simple
computational model—now known as a Turing machine—and argued that any “effective
process” can be formulated as a program for such a machine. (is argument is known

                           6              eval               720

                     (define (factorial n)
                       (if (= n 1)
                           (* (factorial (- n 1)) n)))

        Figure 4.3: e evaluator emulating a factorial machine.

electrical circuits. is would be a circuit that takes as input a signal
encoding the plans for some other circuit, such as a filter. Given this in-
put, the circuit evaluator would then behave like a filter with the same
description. Such a universal electrical circuit is almost unimaginably
complex. It is remarkable that the program evaluator is a rather simple
as the Church-Turing thesis.) Turing then implemented a universal machine, i.e., a Tur-
ing machine that behaves as an evaluator for Turing-machine programs. He used this
framework to demonstrate that there are well-posed problems that cannot be computed
by Turing machines (see Exercise 4.15), and so by implication cannot be formulated as
“effective processes.” Turing went on to make fundamental contributions to practical
computer science as well. For example, he invented the idea of structuring programs
using general-purpose subroutines. See Hodges 1983 for a biography of Turing.
   20 Some people find it counterintuitive that an evaluator, which is implemented by

a relatively simple procedure, can emulate programs that are more complex than the
evaluator itself. e existence of a universal evaluator machine is a deep and wonderful
property of computation. Recursion theory, a branch of mathematical logic, is concerned
with logical limits of computation. Douglas Hofstadter’s beautiful book Gödel, Escher,
Bach explores some of these ideas (Hofstadter 1979).

     Another striking aspect of the evaluator is that it acts as a bridge
between the data objects that are manipulated by our programming lan-
guage and the programming language itself. Imagine that the evaluator
program (implemented in Lisp) is running, and that a user is typing ex-
pressions to the evaluator and observing the results. From the perspec-
tive of the user, an input expression such as (* x x) is an expression in
the programming language, which the evaluator should execute. From
the perspective of the evaluator, however, the expression is simply a list
(in this case, a list of three symbols: *, x, and x) that is to be manipulated
according to a well-defined set of rules.
     at the user’s programs are the evaluator’s data need not be a
source of confusion. In fact, it is sometimes convenient to ignore this
distinction, and to give the user the ability to explicitly evaluate a data
object as a Lisp expression, by making eval available for use in pro-
grams. Many Lisp dialects provide a primitive eval procedure that takes
as arguments an expression and an environment and evaluates the ex-
pression relative to the environment.21 us,
(eval '(* 5 5) user-initial-environment)

(eval (cons '* (list 5 5)) user-initial-environment)

will both return 25.22
   21 Warning: is eval primitive is not identical to the eval procedure we imple-

mented in Section 4.1.1, because it uses actual Scheme environments rather than the
sample environment structures we built in Section 4.1.3. ese actual environments
cannot be manipulated by the user as ordinary lists; they must be accessed via eval or
other special operations. Similarly, the apply primitive we saw earlier is not identical
to the metacircular apply, because it uses actual Scheme procedures rather than the
procedure objects we constructed in Section 4.1.3 and Section 4.1.4.
   22 e  implementation of Scheme includes eval, as well as a symbol user-

initial-environment that is bound to the initial environment in which the user’s in-

       Exercise 4.15: Given a one-argument procedure p and an
       object a, p is said to “halt” on a if evaluating the expres-
       sion (p a) returns a value (as opposed to terminating with
       an error message or running forever). Show that it is im-
       possible to write a procedure halts? that correctly deter-
       mines whether p halts on a for any procedure p and object
       a. Use the following reasoning: If you had such a procedure
       halts?, you could implement the following program:

       (define (run-forever) (run-forever))
       (define (try p)
          (if (halts? p p) (run-forever) 'halted))

       Now consider evaluating the expression (try try) and
       show that any possible outcome (either halting or running
       forever) violates the intended behavior of halts?.23

4.1.6 Internal Definitions
Our environment model of evaluation and our metacircular evaluator
execute definitions in sequence, extending the environment frame one
definition at a time. is is particularly convenient for interactive pro-
gram development, in which the programmer needs to freely mix the
application of procedures with the definition of new procedures. How-
ever, if we think carefully about the internal definitions used to im-
plement block structure (introduced in Section 1.1.8), we will find that
put expressions are evaluated.
   23 Although we stipulated that halts? is given a procedure object, notice that this

reasoning still applies even if halts? can gain access to the procedure’s text and its
environment. is is Turing’s celebrated Halting eorem, which gave the first clear
example of a non-computable problem, i.e., a well-posed task that cannot be carried out
as a computational procedure.

name-by-name extension of the environment may not be the best way
to define local variables.
    Consider a procedure with internal definitions, such as
(define (f x)
  (define (even? n) (if (= n 0) true        (odd?   (- n 1))))
  (define (odd? n)     (if (= n 0) false (even? (- n 1))))
  ⟨rest of body of f⟩)

Our intention here is that the name odd? in the body of the procedure
even? should refer to the procedure odd? that is defined aer even?.
e scope of the name odd? is the entire body of f, not just the portion
of the body of f starting at the point where the define for odd? occurs.
Indeed, when we consider that odd? is itself defined in terms of even?—
so that even? and odd? are mutually recursive procedures—we see that
the only satisfactory interpretation of the two defines is to regard them
as if the names even? and odd? were being added to the environment
simultaneously. More generally, in block structure, the scope of a local
name is the entire procedure body in which the define is evaluated.
     As it happens, our interpreter will evaluate calls to f correctly, but
for an “accidental” reason: Since the definitions of the internal proce-
dures come first, no calls to these procedures will be evaluated until
all of them have been defined. Hence, odd? will have been defined by
the time even? is executed. In fact, our sequential evaluation mecha-
nism will give the same result as a mechanism that directly implements
simultaneous definition for any procedure in which the internal defini-
tions come first in a body and evaluation of the value expressions for
the defined variables doesn’t actually use any of the defined variables.
(For an example of a procedure that doesn’t obey these restrictions, so
that sequential definition isn’t equivalent to simultaneous definition,

see Exercise 4.19.)24
    ere is, however, a simple way to treat definitions so that inter-
nally defined names have truly simultaneous scope—just create all local
variables that will be in the current environment before evaluating any
of the value expressions. One way to do this is by a syntax transfor-
mation on lambda expressions. Before evaluating the body of a lambda
expression, we “scan out” and eliminate all the internal definitions in
the body. e internally defined variables will be created with a let
and then set to their values by assignment. For example, the procedure
(lambda ⟨vars⟩
  (define u ⟨e1⟩)
  (define v ⟨e2⟩)

would be transformed into
(lambda   ⟨vars⟩
  (let ((u '*unassigned*)
          (v '*unassigned*))
     (set! u   ⟨e1⟩)
     (set! v   ⟨e2⟩)

  24 Wanting   programs to not depend on this evaluation mechanism is the reason for
the “management is not responsible” remark in Footnote 28 of Chapter 1. By insisting
that internal definitions come first and do not use each other while the definitions are
being evaluated, the  standard for Scheme leaves implementors some choice in
the mechanism used to evaluate these definitions. e choice of one evaluation rule
rather than another here may seem like a small issue, affecting only the interpretation
of “badly formed” programs. However, we will see in Section 5.5.6 that moving to a
model of simultaneous scoping for internal definitions avoids some nasty difficulties
that would otherwise arise in implementing a compiler.

where *unassigned* is a special symbol that causes looking up a vari-
able to signal an error if an aempt is made to use the value of the
not-yet-assigned variable.
    An alternative strategy for scanning out internal definitions is shown
in Exercise 4.18. Unlike the transformation shown above, this enforces
the restriction that the defined variables’ values can be evaluated with-
out using any of the variables’ values.25

       Exercise 4.16: In this exercise we implement the method
       just described for interpreting internal definitions. We as-
       sume that the evaluator supports let (see Exercise 4.6).

          a. Change lookup-variable-value (Section 4.1.3) to sig-
             nal an error if the value it finds is the symbol *unassigned*.
          b. Write a procedure scan-out-defines that takes a pro-
             cedure body and returns an equivalent one that has
             no internal definitions, by making the transformation
             described above.
          c. Install scan-out-defines in the interpreter, either in
             make-procedure or in procedure-body (see Section
             4.1.3). Which place is beer? Why?

       Exercise 4.17: Draw diagrams of the environment in effect
       when evaluating the expression ⟨e3 ⟩ in the procedure in the
  25 e    standard for Scheme allows for different implementation strategies by
specifying that it is up to the programmer to obey this restriction, not up to the imple-
mentation to enforce it. Some Scheme implementations, including  Scheme, use the
transformation shown above. us, some programs that don’t obey this restriction will
in fact run in such implementations.

text, comparing how this will be structured when defini-
tions are interpreted sequentially with how it will be struc-
tured if definitions are scanned out as described. Why is
there an extra frame in the transformed program? Explain
why this difference in environment structure can never make
a difference in the behavior of a correct program. Design a
way to make the interpreter implement the “simultaneous”
scope rule for internal definitions without constructing the
extra frame.

Exercise 4.18: Consider an alternative strategy for scan-
ning out definitions that translates the example in the text
(lambda   ⟨vars⟩
  (let ((u '*unassigned*) (v '*unassigned*))
    (let ((a   ⟨e1⟩) (b ⟨e2⟩))
       (set! u a)
       (set! v b))

Here a and b are meant to represent new variable names,
created by the interpreter, that do not appear in the user’s
program. Consider the solve procedure from Section 3.5.4:
(define (solve f y0 dt)
  (define    y (integral (delay dy) y0 dt))
  (define dy (stream-map f y))

Will this procedure work if internal definitions are scanned
out as shown in this exercise? What if they are scanned out
as shown in the text? Explain.

       Exercise 4.19: Ben Bitdiddle, Alyssa P. Hacker, and Eva Lu
       Ator are arguing about the desired result of evaluating the
       (let ((a 1))
          (define (f x)
            (define b (+ a x))
            (define a 5)
            (+ a b))
          (f 10))

       Ben asserts that the result should be obtained using the se-
       quential rule for define: b is defined to be 11, then a is de-
       fined to be 5, so the result is 16. Alyssa objects that mutual
       recursion requires the simultaneous scope rule for internal
       procedure definitions, and that it is unreasonable to treat
       procedure names differently from other names. us, she
       argues for the mechanism implemented in Exercise 4.16.
       is would lead to a being unassigned at the time that the
       value for b is to be computed. Hence, in Alyssa’s view the
       procedure should produce an error. Eva has a third opinion.
       She says that if the definitions of a and b are truly meant
       to be simultaneous, then the value 5 for a should be used in
       evaluating b. Hence, in Eva’s view a should be 5, b should be
       15, and the result should be 20. Which (if any) of these view-
       points do you support? Can you devise a way to implement
       internal definitions so that they behave as Eva prefers?26
   26 e  implementors of Scheme support Alyssa on the following grounds: Eva is

in principle correct—the definitions should be regarded as simultaneous. But it seems
difficult to implement a general, efficient mechanism that does what Eva requires. In
the absence of such a mechanism, it is beer to generate an error in the difficult cases
of simultaneous definitions (Alyssa’s notion) than to produce an incorrect answer (as
Ben would have it).
Exercise 4.20: Because internal definitions look sequen-
tial but are actually simultaneous, some people prefer to
avoid them entirely, and use the special form letrec in-
stead. letrec looks like let, so it is not surprising that the
variables it binds are bound simultaneously and have the
same scope as each other. e sample procedure f above
can be wrien without internal definitions, but with ex-
actly the same meaning, as
(define (f x)
    ((even? (lambda (n)
                 (if (= n 0) true       (odd?    (- n 1)))))
     (odd?     (lambda (n)
                 (if (= n 0) false (even? (- n 1))))))
    ⟨rest of body of f⟩))

letrec    expressions, which have the form
(letrec ((⟨var1 ⟩   ⟨exp1 ⟩) . . . (⟨varn ⟩ ⟨expn ⟩))

are a variation on let in which the expressions ⟨expk ⟩ that
provide the initial values for the variables ⟨var k ⟩ are eval-
uated in an environment that includes all the letrec bind-
ings. is permits recursion in the bindings, such as the
mutual recursion of even? and odd? in the example above,
or the evaluation of 10 factorial with
  ((fact (lambda (n)
              (if (= n 1) 1 (* n (fact (- n 1)))))))
  (fact 10))

          a. Implement letrec as a derived expression, by trans-
             forming a letrec expression into a let expression as
             shown in the text above or in Exercise 4.18. at is,
             the letrec variables should be created with a let and
             then be assigned their values with set!.
          b. Louis Reasoner is confused by all this fuss about inter-
             nal definitions. e way he sees it, if you don’t like to
             use define inside a procedure, you can just use let.
             Illustrate what is loose about his reasoning by draw-
             ing an environment diagram that shows the environ-
             ment in which the ⟨rest of body of f ⟩ is evaluated dur-
             ing evaluation of the expression (f 5), with f defined
             as in this exercise. Draw an environment diagram for
             the same evaluation, but with let in place of letrec
             in the definition of f.

       Exercise 4.21: Amazingly, Louis’s intuition in Exercise 4.20
       is correct. It is indeed possible to specify recursive proce-
       dures without using letrec (or even define), although the
       method for accomplishing this is much more subtle than
       Louis imagined. e following expression computes 10 fac-
       torial by applying a recursive factorial procedure:27
       ((lambda (n)
           ((lambda (fact) (fact fact n))
              (lambda (ft k) (if (= k 1) 1 (* k (ft ft (- k 1)))))))

  27 is example illustrates a programming trick for formulating recursive procedures

without using define. e most general trick of this sort is the Y operator, which can be
used to give a “pure λ-calculus” implementation of recursion. (See Stoy 1977 for details
on the λ-calculus, and Gabriel 1988 for an exposition of the Y operator in Scheme.)

        a. Check (by evaluating the expression) that this really
           does compute factorials. Devise an analogous expres-
           sion for computing Fibonacci numbers.
        b. Consider the following procedure, which includes mu-
           tually recursive internal definitions:
           (define (f x)
             (define (even? n)
               (if (= n 0) true    (odd?    (- n 1))))
             (define (odd? n)
               (if (= n 0) false (even? (- n 1))))
             (even? x))

           Fill in the missing expressions to complete an alterna-
           tive definition of f, which uses neither internal defi-
           nitions nor letrec:
           (define (f x)
             ((lambda (even? odd?) (even? even? odd? x))
              (lambda (ev? od? n)
                 (if (= n 0) true (od?   ⟨??⟩ ⟨??⟩ ⟨??⟩)))
              (lambda (ev? od? n)
                 (if (= n 0) false (ev?    ⟨??⟩ ⟨??⟩ ⟨??⟩)))))

4.1.7 Separating Syntactic Analysis from Execution
e evaluator implemented above is simple, but it is very inefficient,
because the syntactic analysis of expressions is interleaved with their
execution. us if a program is executed many times, its syntax is an-
alyzed many times. Consider, for example, evaluating (factorial 4)
using the following definition of factorial:
(define (factorial n)
  (if (= n 1) 1 (* (factorial (- n 1)) n)))

Each time factorial is called, the evaluator must determine that the
body is an if expression and extract the predicate. Only then can it
evaluate the predicate and dispatch on its value. Each time it evaluates
the expression (* (factorial (- n 1)) n), or the subexpressions
(factorial (- n 1)) and (- n 1), the evaluator must perform the
case analysis in eval to determine that the expression is an application,
and must extract its operator and operands. is analysis is expensive.
Performing it repeatedly is wasteful.
    We can transform the evaluator to be significantly more efficient
by arranging things so that syntactic analysis is performed only once.28
We split eval, which takes an expression and an environment, into two
parts. e procedure analyze takes only the expression. It performs the
syntactic analysis and returns a new procedure, the execution procedure,
that encapsulates the work to be done in executing the analyzed expres-
sion. e execution procedure takes an environment as its argument
and completes the evaluation. is saves work because analyze will be
called only once on an expression, while the execution procedure may
be called many times.
    With the separation into analysis and execution, eval now becomes
(define (eval exp env) ((analyze exp) env))

e result of calling analyze is the execution procedure to be applied
to the environment. e analyze procedure is the same case analysis as
performed by the original eval of Section 4.1.1, except that the proce-
dures to which we dispatch perform only analysis, not full evaluation:
(define (analyze exp)

  28 is technique is an integral part of the compilation process, which we shall discuss

in Chapter 5. Jonathan Rees wrote a Scheme interpreter like this in about 1982 for the T
project (Rees and Adams 1982). Marc Feeley (1986) (see also Feeley and Lapalme 1987)
independently invented this technique in his master’s thesis.

  (cond ((self-evaluating? exp) (analyze-self-evaluating exp))
            ((quoted? exp) (analyze-quoted exp))
            ((variable? exp) (analyze-variable exp))
            ((assignment? exp) (analyze-assignment exp))
            ((definition? exp) (analyze-definition exp))
            ((if? exp) (analyze-if exp))
            ((lambda? exp) (analyze-lambda exp))
            ((begin? exp) (analyze-sequence (begin-actions exp)))
            ((cond? exp) (analyze (cond->if exp)))
            ((application? exp) (analyze-application exp))
            (else (error "Unknown expression type: ANALYZE" exp))))

Here is the simplest syntactic analysis procedure, which handles self-
evaluating expressions. It returns an execution procedure that ignores
its environment argument and just returns the expression:
(define (analyze-self-evaluating exp)
  (lambda (env) exp))

For a quoted expression, we can gain a lile efficiency by extracting the
text of the quotation only once, in the analysis phase, rather than in the
execution phase.
(define (analyze-quoted exp)
  (let ((qval (text-of-quotation exp)))
     (lambda (env) qval)))

Looking up a variable value must still be done in the execution phase,
since this depends upon knowing the environment.29
(define (analyze-variable exp)
  (lambda (env) (lookup-variable-value exp env)))

  29 ere   is, however, an important part of the variable search that can be done as
part of the syntactic analysis. As we will show in Section 5.5.6, one can determine the
position in the environment structure where the value of the variable will be found, thus
obviating the need to scan the environment for the entry that matches the variable.
analyze-assignment     also must defer actually seing the variable un-
til the execution, when the environment has been supplied. However,
the fact that the assignment-value expression can be analyzed (re-
cursively) during analysis is a major gain in efficiency, because the
assignment-value expression will now be analyzed only once. e same
holds true for definitions.
(define (analyze-assignment exp)
  (let ((var (assignment-variable exp))
        (vproc (analyze (assignment-value exp))))
    (lambda (env)
      (set-variable-value! var (vproc env) env)
(define (analyze-definition exp)
  (let ((var (definition-variable exp))
        (vproc (analyze (definition-value exp))))
    (lambda (env)
      (define-variable! var (vproc env) env)

For if expressions, we extract and analyze the predicate, consequent,
and alternative at analysis time.
(define (analyze-if exp)
  (let ((pproc (analyze (if-predicate exp)))
        (cproc (analyze (if-consequent exp)))
        (aproc (analyze (if-alternative exp))))
    (lambda (env) (if (true? (pproc env))
                       (cproc env)
                       (aproc env)))))

Analyzing a lambda expression also achieves a major gain in efficiency:
We analyze the lambda body only once, even though procedures result-
ing from evaluation of the lambda may be applied many times.

(define (analyze-lambda exp)
  (let ((vars (lambda-parameters exp))
             (bproc (analyze-sequence (lambda-body exp))))
     (lambda (env) (make-procedure vars bproc env))))

Analysis of a sequence of expressions (as in a begin or the body of a
lambda expression) is more involved.30 Each expression in the sequence
is analyzed, yielding an execution procedure. ese execution proce-
dures are combined to produce an execution procedure that takes an
environment as argument and sequentially calls each individual execu-
tion procedure with the environment as argument.
(define (analyze-sequence exps)
  (define (sequentially proc1 proc2)
     (lambda (env) (proc1 env) (proc2 env)))
  (define (loop first-proc rest-procs)
     (if (null? rest-procs)
             (loop (sequentially first-proc (car rest-procs))
                     (cdr rest-procs))))
  (let ((procs (map analyze exps)))
     (if (null? procs) (error "Empty sequence: ANALYZE"))
     (loop (car procs) (cdr procs))))

To analyze an application, we analyze the operator and operands and
construct an execution procedure that calls the operator execution pro-
cedure (to obtain the actual procedure to be applied) and the operand
execution procedures (to obtain the actual arguments). We then pass
these to execute-application, which is the analog of apply in Section
4.1.1. execute-application differs from apply in that the procedure
body for a compound procedure has already been analyzed, so there is
  30 See   Exercise 4.23 for some insight into the processing of sequences.

no need to do further analysis. Instead, we just call the execution pro-
cedure for the body on the extended environment.
(define (analyze-application exp)
  (let ((fproc (analyze (operator exp)))
        (aprocs (map analyze (operands exp))))
    (lambda (env)
       (fproc env)
       (map (lambda (aproc) (aproc env))
(define (execute-application proc args)
  (cond ((primitive-procedure? proc)
          (apply-primitive-procedure proc args))
        ((compound-procedure? proc)
          ((procedure-body proc)
            (procedure-parameters proc)
            (procedure-environment proc))))
          (error "Unknown procedure type: EXECUTE-APPLICATION"

Our new evaluator uses the same data structures, syntax procedures,
and run-time support procedures as in sections Section 4.1.2, Section
4.1.3, and Section 4.1.4.

      Exercise 4.22: Extend the evaluator in this section to sup-
      port the special form let. (See Exercise 4.6.)

      Exercise 4.23: Alyssa P. Hacker doesn’t understand why
      analyze-sequence needs to be so complicated. All the other
      analysis procedures are straightforward transformations of

the corresponding evaluation procedures (or eval clauses)
in Section 4.1.1. She expected analyze-sequence to look
like this:
(define (analyze-sequence exps)
  (define (execute-sequence procs env)
    (cond ((null? (cdr procs))
            ((car procs) env))
            ((car procs) env)
            (execute-sequence (cdr procs) env))))
  (let ((procs (map analyze exps)))
    (if (null? procs)
         (error "Empty sequence: ANALYZE"))
    (lambda (env)
      (execute-sequence procs env))))

Eva Lu Ator explains to Alyssa that the version in the text
does more of the work of evaluating a sequence at analysis
time. Alyssa’s sequence-execution procedure, rather than
having the calls to the individual execution procedures built
in, loops through the procedures in order to call them: In
effect, although the individual expressions in the sequence
have been analyzed, the sequence itself has not been.
Compare the two versions of analyze-sequence. For ex-
ample, consider the common case (typical of procedure bod-
ies) where the sequence has just one expression. What work
will the execution procedure produced by Alyssa’s program
do? What about the execution procedure produced by the
program in the text above? How do the two versions com-
pare for a sequence with two expressions?

       Exercise 4.24: Design and carry out some experiments to
       compare the speed of the original metacircular evaluator
       with the version in this section. Use your results to esti-
       mate the fraction of time that is spent in analysis versus
       execution for various procedures.

4.2 Variations on a Scheme — Lazy Evaluation
Now that we have an evaluator expressed as a Lisp program, we can
experiment with alternative choices in language design simply by mod-
ifying the evaluator. Indeed, new languages are oen invented by first
writing an evaluator that embeds the new language within an exist-
ing high-level language. For example, if we wish to discuss some aspect
of a proposed modification to Lisp with another member of the Lisp
community, we can supply an evaluator that embodies the change. e
recipient can then experiment with the new evaluator and send back
comments as further modifications. Not only does the high-level imple-
mentation base make it easier to test and debug the evaluator; in addi-
tion, the embedding enables the designer to snarf 31 features from the
underlying language, just as our embedded Lisp evaluator uses primi-
tives and control structure from the underlying Lisp. Only later (if ever)
need the designer go to the trouble of building a complete implemen-
tation in a low-level language or in hardware. In this section and the
next we explore some variations on Scheme that provide significant ad-
ditional expressive power.
  31 Snarf:“To grab, especially a large document or file for the purpose of using it ei-
ther with or without the owner’s permission.” Snarf down: “To snarf, sometimes with
the connotation of absorbing, processing, or understanding.” (ese definitions were
snarfed from Steele et al. 1983. See also Raymond 1993.)

4.2.1 Normal Order and Applicative Order
In Section 1.1, where we began our discussion of models of evaluation,
we noted that Scheme is an applicative-order language, namely, that all
the arguments to Scheme procedures are evaluated when the procedure
is applied. In contrast, normal-order languages delay evaluation of pro-
cedure arguments until the actual argument values are needed. Delay-
ing evaluation of procedure arguments until the last possible moment
(e.g., until they are required by a primitive operation) is called lazy eval-
uation.32 Consider the procedure
(define (try a b) (if (= a 0) 1 b))

Evaluating (try 0 (/ 1 0)) generates an error in Scheme. With lazy
evaluation, there would be no error. Evaluating the expression would
return 1, because the argument (/ 1 0) would never be evaluated.
    An example that exploits lazy evaluation is the definition of a pro-
cedure unless
(define (unless condition usual-value exceptional-value)
  (if condition exceptional-value usual-value))

that can be used in expressions such as
(unless (= b 0)
          (/ a b)
          (begin (display "exception: returning 0") 0))

is won’t work in an applicative-order language because both the usual
value and the exceptional value will be evaluated before unless is called
  32 e  difference between the “lazy” terminology and the “normal-order” terminol-
ogy is somewhat fuzzy. Generally, “lazy” refers to the mechanisms of particular eval-
uators, while “normal-order” refers to the semantics of languages, independent of any
particular evaluation strategy. But this is not a hard-and-fast distinction, and the two
terminologies are oen used interchangeably.

(compare Exercise 1.6). An advantage of lazy evaluation is that some
procedures, such as unless, can do useful computation even if evalu-
ation of some of their arguments would produce errors or would not
     If the body of a procedure is entered before an argument has been
evaluated we say that the procedure is non-strict in that argument. If the
argument is evaluated before the body of the procedure is entered we
say that the procedure is strict in that argument.33 In a purely applicative-
order language, all procedures are strict in each argument. In a purely
normal-order language, all compound procedures are non-strict in each
argument, and primitive procedures may be either strict or non-strict.
ere are also languages (see Exercise 4.31) that give programmers de-
tailed control over the strictness of the procedures they define.
     A striking example of a procedure that can usefully be made non-
strict is cons (or, in general, almost any constructor for data structures).
One can do useful computation, combining elements to form data struc-
tures and operating on the resulting data structures, even if the values
of the elements are not known. It makes perfect sense, for instance, to
compute the length of a list without knowing the values of the indi-
vidual elements in the list. We will exploit this idea in Section 4.2.3 to
implement the streams of Chapter 3 as lists formed of non-strict cons

       Exercise 4.25: Suppose that (in ordinary applicative-order
       Scheme) we define unless as shown above and then define
  33 e   “strict” versus “non-strict” terminology means essentially the same thing as
“applicative-order” versus “normal-order,” except that it refers to individual procedures
and arguments rather than to the language as a whole. At a conference on programming
languages you might hear someone say, “e normal-order language Hassle has certain
strict primitives. Other procedures take their arguments by lazy evaluation.”

      factorial   in terms of unless as
      (define (factorial n)
        (unless (= n 1)
                  (* n (factorial (- n 1)))

      What happens if we aempt to evaluate (factorial 5)?
      Will our definitions work in a normal-order language?

      Exercise 4.26: Ben Bitdiddle and Alyssa P. Hacker disagree
      over the importance of lazy evaluation for implementing
      things such as unless. Ben points out that it’s possible to
      implement unless in applicative order as a special form.
      Alyssa counters that, if one did that, unless would be merely
      syntax, not a procedure that could be used in conjunction
      with higher-order procedures. Fill in the details on both
      sides of the argument. Show how to implement unless as
      a derived expression (like cond or let), and give an exam-
      ple of a situation where it might be useful to have unless
      available as a procedure, rather than as a special form.

4.2.2 An Interpreter with Lazy Evaluation
In this section we will implement a normal-order language that is the
same as Scheme except that compound procedures are non-strict in each
argument. Primitive procedures will still be strict. It is not difficult to
modify the evaluator of Section 4.1.1 so that the language it interprets
behaves this way. Almost all the required changes center around pro-
cedure application.
    e basic idea is that, when applying a procedure, the interpreter
must determine which arguments are to be evaluated and which are to

be delayed. e delayed arguments are not evaluated; instead, they are
transformed into objects called thunks.34 e thunk must contain the
information required to produce the value of the argument when it is
needed, as if it had been evaluated at the time of the application. us,
the thunk must contain the argument expression and the environment
in which the procedure application is being evaluated.
    e process of evaluating the expression in a thunk is called forc-
ing. In general, a thunk will be forced only when its value is needed:
when it is passed to a primitive procedure that will use the value of the
thunk; when it is the value of a predicate of a conditional; and when it is
the value of an operator that is about to be applied as a procedure. One
design choice we have available is whether or not to memoize thunks, as
we did with delayed objects in Section 3.5.1. With memoization, the first
time a thunk is forced, it stores the value that is computed. Subsequent
forcings simply return the stored value without repeating the computa-
tion. We’ll make our interpreter memoize, because this is more efficient
for many applications. ere are tricky considerations here, however.36
  34 e   word thunk was invented by an informal working group that was discussing
the implementation of call-by-name in Algol 60. ey observed that most of the analysis
of (“thinking about”) the expression could be done at compile time; thus, at run time,
the expression would already have been “thunk” about (Ingerman et al. 1960).
   35 is is analogous to the use of force on the delayed objects that were introduced

in Chapter 3 to represent streams. e critical difference between what we are doing
here and what we did in Chapter 3 is that we are building delaying and forcing into the
evaluator, and thus making this uniform and automatic throughout the language.
   36 Lazy evaluation combined with memoization is sometimes referred to as call-by-

need argument passing, in contrast to call-by-name argument passing. (Call-by-name,
introduced in Algol 60, is similar to non-memoized lazy evaluation.) As language de-
signers, we can build our evaluator to memoize, not to memoize, or leave this an option
for programmers (Exercise 4.31). As you might expect from Chapter 3, these choices
raise issues that become both subtle and confusing in the presence of assignments. (See
Exercise 4.27 and Exercise 4.29.) An excellent article by Clinger (1982) aempts to clar-

Modifying the evaluator
e main difference between the lazy evaluator and the one in Section
4.1 is in the handling of procedure applications in eval and apply.
     e application? clause of eval becomes
((application? exp)
 (apply (actual-value (operator exp) env)
          (operands exp)

is is almost the same as the application? clause of eval in Sec-
tion 4.1.1. For lazy evaluation, however, we call apply with the operand
expressions, rather than the arguments produced by evaluating them.
Since we will need the environment to construct thunks if the argu-
ments are to be delayed, we must pass this as well. We still evaluate
the operator, because apply needs the actual procedure to be applied in
order to dispatch on its type (primitive versus compound) and apply it.
    Whenever we need the actual value of an expression, we use
(define (actual-value exp env)
  (force-it (eval exp env)))

instead of just eval, so that if the expression’s value is a thunk, it will
be forced.
    Our new version of apply is also almost the same as the version
in Section 4.1.1. e difference is that eval has passed in unevaluated
operand expressions: For primitive procedures (which are strict), we
evaluate all the arguments before applying the primitive; for compound
procedures (which are non-strict) we delay all the arguments before ap-
plying the procedure.
ify the multiple dimensions of confusion that arise here.

(define (apply procedure arguments env)
  (cond ((primitive-procedure? procedure)
            (list-of-arg-values arguments env)))    ; changed
        ((compound-procedure? procedure)
            (procedure-body procedure)
            (procedure-parameters procedure)
            (list-of-delayed-args arguments env)    ; changed
            (procedure-environment procedure))))
        (else (error "Unknown procedure type: APPLY"

e procedures that process the arguments are just like list-of-values
from Section 4.1.1, except that list-of-delayed-args delays the argu-
ments instead of evaluating them, and list-of-arg-values uses actual-
value instead of eval:

(define (list-of-arg-values exps env)
  (if (no-operands? exps)
      (cons (actual-value (first-operand exps)
             (list-of-arg-values (rest-operands exps)
(define (list-of-delayed-args exps env)
  (if (no-operands? exps)
      (cons (delay-it (first-operand exps)
             (list-of-delayed-args (rest-operands exps)

e other place we must change the evaluator is in the handling of if,
where we must use actual-value instead of eval to get the value of
the predicate expression before testing whether it is true or false:
(define (eval-if exp env)
  (if (true? (actual-value (if-predicate exp) env))
      (eval (if-consequent exp) env)
      (eval (if-alternative exp) env)))

Finally, we must change the driver-loop procedure (Section 4.1.4) to
use actual-value instead of eval, so that if a delayed value is prop-
agated back to the read-eval-print loop, it will be forced before being
printed. We also change the prompts to indicate that this is the lazy
(define input-prompt    ";;; L-Eval input:")
(define output-prompt ";;; L-Eval value:")
(define (driver-loop)
  (prompt-for-input input-prompt)
  (let ((input (read)))
    (let ((output
             input the-global-environment)))
      (announce-output output-prompt)
      (user-print output)))

With these changes made, we can start the evaluator and test it. e
successful evaluation of the try expression discussed in Section 4.2.1
indicates that the interpreter is performing lazy evaluation:
(define the-global-environment (setup-environment))
;;; L-Eval input:
(define (try a b) (if (= a 0) 1 b))

;;; L-Eval value:
;;; L-Eval input:
(try 0 (/ 1 0))
;;; L-Eval value:

Representing thunks
Our evaluator must arrange to create thunks when procedures are ap-
plied to arguments and to force these thunks later. A thunk must pack-
age an expression together with the environment, so that the argument
can be produced later. To force the thunk, we simply extract the ex-
pression and environment from the thunk and evaluate the expression
in the environment. We use actual-value rather than eval so that in
case the value of the expression is itself a thunk, we will force that, and
so on, until we reach something that is not a thunk:
(define (force-it obj)
    (if (thunk? obj)
        (actual-value (thunk-exp obj) (thunk-env obj))

One easy way to package an expression with an environment is to make
a list containing the expression and the environment. us, we create a
thunk as follows:
(define (delay-it exp env)
    (list 'thunk exp env))
(define (thunk? obj)
    (tagged-list? obj 'thunk))
(define (thunk-exp thunk) (cadr       thunk))
(define (thunk-env thunk) (caddr thunk))

Actually, what we want for our interpreter is not quite this, but rather
thunks that have been memoized. When a thunk is forced, we will turn
it into an evaluated thunk by replacing the stored expression with its
value and changing the thunk tag so that it can be recognized as already
(define (evaluated-thunk? obj)
  (tagged-list? obj 'evaluated-thunk))
(define (thunk-value evaluated-thunk)
  (cadr evaluated-thunk))
(define (force-it obj)
  (cond ((thunk? obj)
              (let ((result (actual-value (thunk-exp obj)
                                                    (thunk-env obj))))
               (set-car! obj 'evaluated-thunk)
               (set-car! (cdr obj)
                             result)          ; replace exp with its value
               (set-cdr! (cdr obj)
                             '())             ; forget unneeded env
           ((evaluated-thunk? obj) (thunk-value obj))
           (else obj)))

Notice that the same delay-it procedure works both with and without
  37 Notice  that we also erase the env from the thunk once the expression’s value has
been computed. is makes no difference in the values returned by the interpreter. It
does help save space, however, because removing the reference from the thunk to the
env once it is no longer needed allows this structure to be garbage-collected and its space
recycled, as we will discuss in Section 5.3.
  Similarly, we could have allowed unneeded environments in the memoized delayed
objects of Section 3.5.1 to be garbage-collected, by having memo-proc do something like
(set! proc '()) to discard the procedure proc (which includes the environment in
which the delay was evaluated) aer storing its value.

       Exercise 4.27: Suppose we type in the following definitions
       to the lazy evaluator:
       (define count 0)
       (define (id x) (set! count (+ count 1)) x)

       Give the missing values in the following sequence of inter-
       actions, and explain your answers.38
       (define w (id (id 10)))
       ;;; L-Eval input:
       ;;; L-Eval value:
       ;;; L-Eval input:
       ;;; L-Eval value:
       ;;; L-Eval input:
       ;;; L-Eval value:

       Exercise 4.28: eval uses actual-value rather than eval
       to evaluate the operator before passing it to apply, in or-
       der to force the value of the operator. Give an example that
       demonstrates the need for this forcing.
       Exercise 4.29: Exhibit a program that you would expect
       to run much more slowly without memoization than with
   38 is exercise demonstrates that the interaction between lazy evaluation and side

effects can be very confusing. is is just what you might expect from the discussion
in Chapter 3.

memoization. Also, consider the following interaction, where
the id procedure is defined as in Exercise 4.27 and count
starts at 0:
(define (square x) (* x x))
;;; L-Eval input:
(square (id 10))
;;; L-Eval value:
;;; L-Eval input:
;;; L-Eval value:

Give the responses both when the evaluator memoizes and
when it does not.

Exercise 4.30: Cy D. Fect, a reformed C programmer, is
worried that some side effects may never take place, be-
cause the lazy evaluator doesn’t force the expressions in a
sequence. Since the value of an expression in a sequence
other than the last one is not used (the expression is there
only for its effect, such as assigning to a variable or print-
ing), there can be no subsequent use of this value (e.g., as an
argument to a primitive procedure) that will cause it to be
forced. Cy thus thinks that when evaluating sequences, we
must force all expressions in the sequence except the final
one. He proposes to modify eval-sequence from Section
4.1.1 to use actual-value rather than eval:
(define (eval-sequence exps env)
  (cond ((last-exp? exps) (eval (first-exp exps) env))
         (else (actual-value (first-exp exps) env)

              (eval-sequence (rest-exps exps) env))))

a. Ben Bitdiddle thinks Cy is wrong. He shows Cy the
   for-each procedure described in Exercise 2.23, which
   gives an important example of a sequence with side
  (define (for-each proc items)
       (if (null? items)
          (begin (proc (car items))
                  (for-each proc (cdr items)))))

  He claims that the evaluator in the text (with the orig-
  inal eval-sequence) handles this correctly:
  ;;; L-Eval input:
  (for-each (lambda (x) (newline) (display x))
              (list 57 321 88))
  ;;; L-Eval value:

  Explain why Ben is right about the behavior of for-

b. Cy agrees that Ben is right about the for-each exam-
   ple, but says that that’s not the kind of program he
   was thinking about when he proposed his change to
   eval-sequence. He defines the following two proce-
   dures in the lazy evaluator:

       (define (p1 x)
           (set! x (cons x '(2)))
       (define (p2 x)
           (define (p e)
           (p (set! x (cons x '(2)))))

       What are the values of (p1 1) and (p2 1) with the
       original eval-sequence? What would the values be
       with Cy’s proposed change to eval-sequence?
  c. Cy also points out that changing eval-sequence as he
     proposes does not affect the behavior of the example
     in part a. Explain why this is true.
  d. How do you think sequences ought to be treated in
     the lazy evaluator? Do you like Cy’s approach, the ap-
     proach in the text, or some other approach?

Exercise 4.31: e approach taken in this section is some-
what unpleasant, because it makes an incompatible change
to Scheme. It might be nicer to implement lazy evaluation
as an upward-compatible extension, that is, so that ordinary
Scheme programs will work as before. We can do this by
extending the syntax of procedure declarations to let the
user control whether or not arguments are to be delayed.
While we’re at it, we may as well also give the user the
choice between delaying with and without memoization.
For example, the definition
(define (f a (b lazy) c (d lazy-memo))
  . . .)

      would define f to be a procedure of four arguments, where
      the first and third arguments are evaluated when the pro-
      cedure is called, the second argument is delayed, and the
      fourth argument is both delayed and memoized. us, or-
      dinary procedure definitions will produce the same behav-
      ior as ordinary Scheme, while adding the lazy-memo dec-
      laration to each parameter of every compound procedure
      will produce the behavior of the lazy evaluator defined in
      this section. Design and implement the changes required
      to produce such an extension to Scheme. You will have to
      implement new syntax procedures to handle the new syn-
      tax for define. You must also arrange for eval or apply to
      determine when arguments are to be delayed, and to force
      or delay arguments accordingly, and you must arrange for
      forcing to memoize or not, as appropriate.

4.2.3 Streams as Lazy Lists
In Section 3.5.1, we showed how to implement streams as delayed lists.
We introduced special forms delay and cons-stream, which allowed
us to construct a “promise” to compute the cdr of a stream, without
actually fulfilling that promise until later. We could use this general
technique of introducing special forms whenever we need more control
over the evaluation process, but this is awkward. For one thing, a spe-
cial form is not a first-class object like a procedure, so we cannot use it
together with higher-order procedures.39 Additionally, we were forced
to create streams as a new kind of data object similar but not identical to
lists, and this required us to reimplement many ordinary list operations
  39 is   is precisely the issue with the unless procedure, as in Exercise 4.26.

(map, append, and so on) for use with streams.
     With lazy evaluation, streams and lists can be identical, so there is
no need for special forms or for separate list and stream operations. All
we need to do is to arrange maers so that cons is non-strict. One way
to accomplish this is to extend the lazy evaluator to allow for non-strict
primitives, and to implement cons as one of these. An easier way is to re-
call (Section 2.1.3) that there is no fundamental need to implement cons
as a primitive at all. Instead, we can represent pairs as procedures:40
(define (cons x y) (lambda (m) (m x y)))
(define (car z) (z (lambda (p q) p)))
(define (cdr z) (z (lambda (p q) q)))

In terms of these basic operations, the standard definitions of the list
operations will work with infinite lists (streams) as well as finite ones,
and the stream operations can be implemented as list operations. Here
are some examples:
(define (list-ref items n)
  (if (= n 0)
        (car items)
        (list-ref (cdr items) (- n 1))))
(define (map proc items)
  (if (null? items)
        (cons (proc (car items)) (map proc (cdr items)))))
(define (scale-list items factor)
  (map (lambda (x) (* x factor)) items))

  40 is is the procedural representation described in Exercise 2.4. Essentially any pro-

cedural representation (e.g., a message-passing implementation) would do as well. No-
tice that we can install these definitions in the lazy evaluator simply by typing them
at the driver loop. If we had originally included cons, car, and cdr as primitives in the
global environment, they will be redefined. (Also see Exercise 4.33 and Exercise 4.34.)

(define (add-lists list1 list2)
  (cond ((null? list1) list2)
           ((null? list2) list1)
           (else (cons (+ (car list1) (car list2))
                           (add-lists (cdr list1) (cdr list2))))))
(define ones (cons 1 ones))
(define integers (cons 1 (add-lists ones integers)))
;;; L-Eval input:
(list-ref integers 17)
;;; L-Eval value:

Note that these lazy lists are even lazier than the streams of Chapter 3:
e car of the list, as well as the cdr, is delayed.41 In fact, even accessing
the car or cdr of a lazy pair need not force the value of a list element.
e value will be forced only when it is really needed—e.g., for use as
the argument of a primitive, or to be printed as an answer.
    Lazy pairs also help with the problem that arose with streams in Sec-
tion 3.5.4, where we found that formulating stream models of systems
with loops may require us to sprinkle our programs with explicit delay
operations, beyond the ones supplied by cons-stream. With lazy evalu-
ation, all arguments to procedures are delayed uniformly. For instance,
we can implement procedures to integrate lists and solve differential
equations as we originally intended in Section 3.5.4:
(define (integral integrand initial-value dt)
  (define int
     (cons initial-value
             (add-lists (scale-list integrand dt) int)))

  41 is  permits us to create delayed versions of more general kinds of list structures,
not just sequences. Hughes 1990 discusses some applications of “lazy trees.”

(define (solve f y0 dt)
 (define    y (integral dy y0 dt))
 (define dy (map f y))
;;; L-Eval input:
(list-ref (solve (lambda (x) x) 1 0.001) 1000)
;;; L-Eval value:

      Exercise 4.32: Give some examples that illustrate the dif-
      ference between the streams of Chapter 3 and the “lazier”
      lazy lists described in this section. How can you take ad-
      vantage of this extra laziness?

      Exercise 4.33: Ben Bitdiddle tests the lazy list implemen-
      tation given above by evaluating the expression:
      (car '(a b c))

      To his surprise, this produces an error. Aer some thought,
      he realizes that the “lists” obtained by reading in quoted
      expressions are different from the lists manipulated by the
      new definitions of cons, car, and cdr. Modify the evalua-
      tor’s treatment of quoted expressions so that quoted lists
      typed at the driver loop will produce true lazy lists.

      Exercise 4.34: Modify the driver loop for the evaluator so
      that lazy pairs and lists will print in some reasonable way.
      (What are you going to do about infinite lists?) You may
      also need to modify the representation of lazy pairs so that
      the evaluator can identify them in order to print them.

4.3 Variations on a Scheme — Nondeterministic
In this section, we extend the Scheme evaluator to support a program-
ming paradigm called nondeterministic computing by building into the
evaluator a facility to support automatic search. is is a much more
profound change to the language than the introduction of lazy evalua-
tion in Section 4.2.
     Nondeterministic computing, like stream processing, is useful for
“generate and test” applications. Consider the task of starting with two
lists of positive integers and finding a pair of integers—one from the first
list and one from the second list—whose sum is prime. We saw how to
handle this with finite sequence operations in Section 2.2.3 and with
infinite streams in Section 3.5.3. Our approach was to generate the se-
quence of all possible pairs and filter these to select the pairs whose sum
is prime. Whether we actually generate the entire sequence of pairs first
as in Chapter 2, or interleave the generating and filtering as in Chap-
ter 3, is immaterial to the essential image of how the computation is
     e nondeterministic approach evokes a different image. Imagine
simply that we choose (in some way) a number from the first list and a
number from the second list and require (using some mechanism) that
their sum be prime. is is expressed by following procedure:
(define (prime-sum-pair list1 list2)
  (let ((a (an-element-of list1))
         (b (an-element-of list2)))
    (require (prime? (+ a b)))
    (list a b)))

It might seem as if this procedure merely restates the problem, rather
than specifying a way to solve it. Nevertheless, this is a legitimate non-
deterministic program.42
     e key idea here is that expressions in a nondeterministic language
can have more than one possible value. For instance, an-element-of
might return any element of the given list. Our nondeterministic pro-
gram evaluator will work by automatically choosing a possible value
and keeping track of the choice. If a subsequent requirement is not met,
the evaluator will try a different choice, and it will keep trying new
choices until the evaluation succeeds, or until we run out of choices.
Just as the lazy evaluator freed the programmer from the details of how
values are delayed and forced, the nondeterministic program evaluator
will free the programmer from the details of how choices are made.
     It is instructive to contrast the different images of time evoked by
nondeterministic evaluation and stream processing. Stream processing
uses lazy evaluation to decouple the time when the stream of possible
answers is assembled from the time when the actual stream elements are
produced. e evaluator supports the illusion that all the possible an-
swers are laid out before us in a timeless sequence. With nondetermin-
istic evaluation, an expression represents the exploration of a set of pos-
sible worlds, each determined by a set of choices. Some of the possible
worlds lead to dead ends, while others have useful values. e nonde-
terministic program evaluator supports the illusion that time branches,
and that our programs have different possible execution histories. When
  42 We assume that we have previously defined a procedure prime? that tests whether

numbers are prime. Even with prime? defined, the prime-sum-pair procedure may
look suspiciously like the unhelpful “pseudo-Lisp” aempt to define the square-root
function, which we described at the beginning of Section 1.1.7. In fact, a square-root
procedure along those lines can actually be formulated as a nondeterministic program.
By incorporating a search mechanism into the evaluator, we are eroding the distinc-
tion between purely declarative descriptions and imperative specifications of how to
compute answers. We’ll go even farther in this direction in Section 4.4.

we reach a dead end, we can revisit a previous choice point and proceed
along a different branch.
    e nondeterministic program evaluator implemented below is called
the amb evaluator because it is based on a new special form called amb.
We can type the above definition of prime-sum-pair at the amb evalu-
ator driver loop (along with definitions of prime?, an-element-of, and
require) and run the procedure as follows:

;;; Amb-Eval input:
(prime-sum-pair '(1 3 5 8) '(20 35 110))
;;; Starting a new problem
;;; Amb-Eval value:
(3 20)

e value returned was obtained aer the evaluator repeatedly chose
elements from each of the lists, until a successful choice was made.
    Section 4.3.1 introduces amb and explains how it supports nondeter-
minism through the evaluator’s automatic search mechanism. Section
4.3.2 presents examples of nondeterministic programs, and Section 4.3.3
gives the details of how to implement the amb evaluator by modifying
the ordinary Scheme evaluator.

4.3.1 Amb and Search
To extend Scheme to support nondeterminism, we introduce a new spe-
cial form called amb.43 e expression
(amb   ⟨e1 ⟩ ⟨e2 ⟩ . . . ⟨en ⟩)

returns the value of one of the n expressions ⟨ei ⟩ “ambiguously.” For
example, the expression
  43 eidea of amb for nondeterministic programming was first described in 1961 by
John McCarthy (see McCarthy 1963).

(list (amb 1 2 3) (amb 'a 'b))

can have six possible values:
(1 a) (1 b) (2 a) (2 b) (3 a) (3 b)

amb  with a single choice produces an ordinary (single) value.
      ambwith no choices—the expression (amb)—is an expression with
no acceptable values. Operationally, we can think of (amb) as an expres-
sion that when evaluated causes the computation to “fail”: e compu-
tation aborts and no value is produced. Using this idea, we can express
the requirement that a particular predicate expression p must be true as
(define (require p) (if (not p) (amb)))

With amb and require, we can implement the an-element-of proce-
dure used above:
(define (an-element-of items)
  (require (not (null? items)))
  (amb (car items) (an-element-of (cdr items))))

an-element-of     fails if the list is empty. Otherwise it ambiguously re-
turns either the first element of the list or an element chosen from the
rest of the list.
    We can also express infinite ranges of choices. e following proce-
dure potentially returns any integer greater than or equal to some given
(define (an-integer-starting-from n)
  (amb n (an-integer-starting-from (+ n 1))))

is is like the stream procedure integers-starting-from described
in Section 3.5.2, but with an important difference: e stream procedure

returns an object that represents the sequence of all integers beginning
with n, whereas the amb procedure returns a single integer.44
     Abstractly, we can imagine that evaluating an amb expression causes
time to split into branches, where the computation continues on each
branch with one of the possible values of the expression. We say that
amb represents a nondeterministic choice point. If we had a machine with a
sufficient number of processors that could be dynamically allocated, we
could implement the search in a straightforward way. Execution would
proceed as in a sequential machine, until an amb expression is encoun-
tered. At this point, more processors would be allocated and initialized
to continue all of the parallel executions implied by the choice. Each
processor would proceed sequentially as if it were the only choice, until
it either terminates by encountering a failure, or it further subdivides,
or it finishes.45
     On the other hand, if we have a machine that can execute only one
process (or a few concurrent processes), we must consider the alterna-
tives sequentially. One could imagine modifying an evaluator to pick
at random a branch to follow whenever it encounters a choice point.
  44 In actuality, the distinction between nondeterministically returning a single choice

and returning all choices depends somewhat on our point of view. From the perspective
of the code that uses the value, the nondeterministic choice returns a single value. From
the perspective of the programmer designing the code, the nondeterministic choice
potentially returns all possible values, and the computation branches so that each value
is investigated separately.
   45 One might object that this is a hopelessly inefficient mechanism. It might require

millions of processors to solve some easily stated problem this way, and most of the
time most of those processors would be idle. is objection should be taken in the
context of history. Memory used to be considered just such an expensive commodity.
In 1964 a megabyte of  cost about $400,000. Now every personal computer has
many megabytes of , and most of the time most of that  is unused. It is hard
to underestimate the cost of mass-produced electronics.

Random choice, however, can easily lead to failing values. We might
try running the evaluator over and over, making random choices and
hoping to find a non-failing value, but it is beer to systematically search
all possible execution paths. e amb evaluator that we will develop and
work with in this section implements a systematic search as follows:
When the evaluator encounters an application of amb, it initially selects
the first alternative. is selection may itself lead to a further choice. e
evaluator will always initially choose the first alternative at each choice
point. If a choice results in a failure, then the evaluator automagically46
backtracks to the most recent choice point and tries the next alternative.
If it runs out of alternatives at any choice point, the evaluator will back
up to the previous choice point and resume from there. is process
leads to a search strategy known as depth-first search or chronological
   46 Automagically: “Automatically, but in a way which, for some reason (typically be-

cause it is too complicated, or too ugly, or perhaps even too trivial), the speaker doesn’t
feel like explaining.” (Steele et al. 1983, Raymond 1993)
   47 e integration of automatic search strategies into programming languages has

had a long and checkered history. e first suggestions that nondeterministic algo-
rithms might be elegantly encoded in a programming language with search and au-
tomatic backtracking came from Robert Floyd (1967). Carl Hewi (1969) invented a
programming language called Planner that explicitly supported automatic chronolog-
ical backtracking, providing for a built-in depth-first search strategy. Sussman et al.
(1971) implemented a subset of this language, called MicroPlanner, which was used
to support work in problem solving and robot planning. Similar ideas, arising from
logic and theorem proving, led to the genesis in Edinburgh and Marseille of the ele-
gant language Prolog (which we will discuss in Section 4.4). Aer sufficient frustration
with automatic search, McDermo and Sussman (1972) developed a language called
Conniver, which included mechanisms for placing the search strategy under program-
mer control. is proved unwieldy, however, and Sussman and Stallman 1975 found a
more tractable approach while investigating methods of symbolic analysis for electrical
circuits. ey developed a non-chronological backtracking scheme that was based on

Driver loop
e driver loop for the amb evaluator has some unusual properties. It
reads an expression and prints the value of the first non-failing execu-
tion, as in the prime-sum-pair example shown above. If we want to see
the value of the next successful execution, we can ask the interpreter to
backtrack and aempt to generate a second non-failing execution. is
is signaled by typing the symbol try-again. If any expression except
try-again is given, the interpreter will start a new problem, discarding
the unexplored alternatives in the previous problem. Here is a sample
;;; Amb-Eval input:
(prime-sum-pair '(1 3 5 8) '(20 35 110))
;;; Starting a new problem
;;; Amb-Eval value:
(3 20)

;;; Amb-Eval input:
;;; Amb-Eval value:

tracing out the logical dependencies connecting facts, a technique that has come to be
known as dependency-directed backtracking. Although their method was complex, it pro-
duced reasonably efficient programs because it did lile redundant search. Doyle (1979)
and McAllester (1978; 1980) generalized and clarified the methods of Stallman and Suss-
man, developing a new paradigm for formulating search that is now called truth main-
tenance. Modern problem-solving systems all use some form of truth-maintenance sys-
tem as a substrate. See Forbus and deKleer 1993 for a discussion of elegant ways to
build truth-maintenance systems and applications using truth maintenance. Zabih et
al. 1987 describes a nondeterministic extension to Scheme that is based on amb; it is
similar to the interpreter described in this section, but more sophisticated, because it
uses dependency-directed backtracking rather than chronological backtracking. Win-
ston 1992 gives an introduction to both kinds of backtracking.

(3 110)

;;; Amb-Eval input:
;;; Amb-Eval value:
(8 35)

;;; Amb-Eval input:
;;; There are no more values of
(prime-sum-pair (quote (1 3 5 8)) (quote (20 35 110)))

;;; Amb-Eval input:
(prime-sum-pair '(19 27 30) '(11 36 58))
;;; Starting a new problem
;;; Amb-Eval value:
(30 11)

     Exercise 4.35: Write a procedure an-integer-between that
     returns an integer between two given bounds. is can be
     used to implement a procedure that finds Pythagorean triples,
     i.e., triples of integers (i, j, k) between the given bounds
     such that i ≤ j and i 2 + j 2 = k 2 , as follows:
     (define (a-pythagorean-triple-between low high)
          (let ((i (an-integer-between low high)))
            (let ((j (an-integer-between i high)))
             (let ((k (an-integer-between j high)))
               (require (= (+ (* i i) (* j j)) (* k k)))
               (list i j k)))))

     Exercise 4.36: Exercise 3.69 discussed how to generate the
     stream of all Pythagorean triples, with no upper bound on

      the size of the integers to be searched. Explain why simply
      replacing an-integer-between by an-integer-starting-
      from in the procedure in Exercise 4.35 is not an adequate
      way to generate arbitrary Pythagorean triples. Write a pro-
      cedure that actually will accomplish this. (at is, write a
      procedure for which repeatedly typing try-again would in
      principle eventually generate all Pythagorean triples.)

      Exercise 4.37: Ben Bitdiddle claims that the following method
      for generating Pythagorean triples is more efficient than
      the one in Exercise 4.35. Is he correct? (Hint: Consider the
      number of possibilities that must be explored.)
      (define (a-pythagorean-triple-between low high)
        (let ((i (an-integer-between low high))
              (hsq (* high high)))
          (let ((j (an-integer-between i high)))
            (let ((ksq (+ (* i i) (* j j))))
              (require (>= hsq ksq))
              (let ((k (sqrt ksq)))
                 (require (integer? k))
                 (list i j k))))))

4.3.2 Examples of Nondeterministic Programs
Section 4.3.3 describes the implementation of the amb evaluator. First,
however, we give some examples of how it can be used. e advantage
of nondeterministic programming is that we can suppress the details of
how search is carried out, thereby expressing our programs at a higher
level of abstraction.

Logic Puzzles
e following puzzle (taken from Dinesman 1968) is typical of a large
class of simple logic puzzles:

         Baker, Cooper, Fletcher, Miller, and Smith live on differ-
         ent floors of an apartment house that contains only five
         floors. Baker does not live on the top floor. Cooper does
         not live on the boom floor. Fletcher does not live on ei-
         ther the top or the boom floor. Miller lives on a higher
         floor than does Cooper. Smith does not live on a floor adja-
         cent to Fletcher’s. Fletcher does not live on a floor adjacent
         to Cooper’s. Where does everyone live?

We can determine who lives on each floor in a straightforward way by
enumerating all the possibilities and imposing the given restrictions:48
(define (multiple-dwelling)
  (let ((baker            (amb 1 2 3 4 5)) (cooper (amb 1 2 3 4 5))
            (fletcher (amb 1 2 3 4 5)) (miller (amb 1 2 3 4 5))
            (smith        (amb 1 2 3 4 5)))
      (distinct? (list baker cooper fletcher miller smith)))
     (require (not (= baker 5)))

  48 Our  program uses the following procedure to determine if the elements of a list
are distinct:
(define (distinct? items)
  (cond ((null? items) true)
           ((null? (cdr items)) true)
           ((member (car items) (cdr items)) false)
           (else (distinct? (cdr items)))))

member   is like memq except that it uses equal? instead of eq? to test for equality.

    (require (not (= cooper 1)))
    (require (not (= fletcher 5)))
    (require (not (= fletcher 1)))
    (require (> miller cooper))
    (require (not (= (abs (- smith fletcher)) 1)))
    (require (not (= (abs (- fletcher cooper)) 1)))
    (list (list 'baker baker)            (list 'cooper cooper)
           (list 'fletcher fletcher) (list 'miller miller)
           (list 'smith smith))))

Evaluating the expression (multiple-dwelling) produces the result
((baker 3) (cooper 2) (fletcher 4) (miller 5) (smith 1))

Although this simple procedure works, it is very slow. Exercise 4.39 and
Exercise 4.40 discuss some possible improvements.

      Exercise 4.38: Modify the multiple-dwelling procedure to
      omit the requirement that Smith and Fletcher do not live
      on adjacent floors. How many solutions are there to this
      modified puzzle?

      Exercise 4.39: Does the order of the restrictions in the multiple-
      dwelling procedure affect the answer? Does it affect the
      time to find an answer? If you think it maers, demonstrate
      a faster program obtained from the given one by reordering
      the restrictions. If you think it does not maer, argue your

      Exercise 4.40: In the multiple dwelling problem, how many
      sets of assignments are there of people to floors, both be-
      fore and aer the requirement that floor assignments be
      distinct? It is very inefficient to generate all possible assign-
      ments of people to floors and then leave it to backtracking

to eliminate them. For example, most of the restrictions de-
pend on only one or two of the person-floor variables, and
can thus be imposed before floors have been selected for
all the people. Write and demonstrate a much more effi-
cient nondeterministic procedure that solves this problem
based upon generating only those possibilities that are not
already ruled out by previous restrictions. (Hint: is will
require a nest of let expressions.)

Exercise 4.41: Write an ordinary Scheme program to solve
the multiple dwelling puzzle.

Exercise 4.42: Solve the following “Liars” puzzle (from Phillips
Five schoolgirls sat for an examination. eir parents—so
they thought—showed an undue degree of interest in the
result. ey therefore agreed that, in writing home about
the examination, each girl should make one true statement
and one untrue one. e following are the relevant passages
from their leers:

   • Bey: “Kiy was second in the examination. I was
     only third.”
   • Ethel: “You’ll be glad to hear that I was on top. Joan
     was 2nd.”
   • Joan: “I was third, and poor old Ethel was boom.”
   • Kiy: “I came out second. Mary was only fourth.”
   • Mary: “I was fourth. Top place was taken by Bey.”

       What in fact was the order in which the five girls were

       Exercise 4.43: Use the amb evaluator to solve the following
       Mary Ann Moore’s father has a yacht and so has each of
       his four friends: Colonel Downing, Mr. Hall, Sir Barnacle
       Hood, and Dr. Parker. Each of the five also has one daugh-
       ter and each has named his yacht aer a daughter of one of
       the others. Sir Barnacle’s yacht is the Gabrielle, Mr. Moore
       owns the Lorna; Mr. Hall the Rosalind. e Melissa, owned
       by Colonel Downing, is named aer Sir Barnacle’s daugh-
       ter. Gabrielle’s father owns the yacht that is named aer
       Dr. Parker’s daughter. Who is Lorna’s father?
       Try to write the program so that it runs efficiently (see Ex-
       ercise 4.40). Also determine how many solutions there are
       if we are not told that Mary Ann’s last name is Moore.

       Exercise 4.44: Exercise 2.42 described the “eight-queens
       puzzle” of placing queens on a chessboard so that no two at-
       tack each other. Write a nondeterministic program to solve
       this puzzle.

Parsing natural language
Programs designed to accept natural language as input usually start by
aempting to parse the input, that is, to match the input against some
grammatical structure. For example, we might try to recognize simple
  49 is is taken from a booklet called “Problematical Recreations,” published in the

1960s by Lion Industries, where it is aributed to the Kansas State Engineer.

sentences consisting of an article followed by a noun followed by a verb,
such as “e cat eats.” To accomplish such an analysis, we must be able
to identify the parts of speech of individual words. We could start with
some lists that classify various words:50
(define nouns '(noun student professor cat class))
(define verbs '(verb studies lectures eats sleeps))
(define articles '(article the a))

We also need a grammar, that is, a set of rules describing how gram-
matical elements are composed from simpler elements. A very simple
grammar might stipulate that a sentence always consists of two pieces—
a noun phrase followed by a verb—and that a noun phrase consists of
an article followed by a noun. With this grammar, the sentence “e cat
eats” is parsed as follows:
(sentence (noun-phrase (article the) (noun cat))
             (verb eats))

We can generate such a parse with a simple program that has separate
procedures for each of the grammatical rules. To parse a sentence, we
identify its two constituent pieces and return a list of these two ele-
ments, tagged with the symbol sentence:
(define (parse-sentence)
  (list 'sentence
            (parse-word verbs)))

A noun phrase, similarly, is parsed by finding an article followed by a
  50 Here we use the convention that the first element of each list designates the part
of speech for the rest of the words in the list.

(define (parse-noun-phrase)
  (list 'noun-phrase
          (parse-word articles)
          (parse-word nouns)))

At the lowest level, parsing boils down to repeatedly checking that the
next unparsed word is a member of the list of words for the required part
of speech. To implement this, we maintain a global variable *unparsed*,
which is the input that has not yet been parsed. Each time we check a
word, we require that *unparsed* must be non-empty and that it should
begin with a word from the designated list. If so, we remove that word
from *unparsed* and return the word together with its part of speech
(which is found at the head of the list):51
(define (parse-word word-list)
  (require (not (null? *unparsed*)))
  (require (memq (car *unparsed*) (cdr word-list)))
  (let ((found-word (car *unparsed*)))
     (set! *unparsed* (cdr *unparsed*))
     (list (car word-list) found-word)))

To start the parsing, all we need to do is set *unparsed* to be the entire
input, try to parse a sentence, and check that nothing is le over:
(define *unparsed* '())
(define (parse input)
  (set! *unparsed* input)
  (let ((sent (parse-sentence)))
     (require (null? *unparsed*)) sent))

We can now try the parser and verify that it works for our simple test
  51 Noticethat parse-word uses set! to modify the unparsed input list. For this to
work, our amb evaluator must undo the effects of set! operations when it backtracks.

;;; Amb-Eval input:
(parse '(the cat eats))
;;; Starting a new problem
;;; Amb-Eval value:

(sentence (noun-phrase (article the) (noun cat)) (verb eats))

e amb evaluator is useful here because it is convenient to express
the parsing constraints with the aid of require. Automatic search and
backtracking really pay off, however, when we consider more complex
grammars where there are choices for how the units can be decom-
    Let’s add to our grammar a list of prepositions:
(define prepositions '(prep for to in by with))

and define a prepositional phrase (e.g., “for the cat”) to be a preposition
followed by a noun phrase:
(define (parse-prepositional-phrase)
  (list 'prep-phrase
          (parse-word prepositions)

Now we can define a sentence to be a noun phrase followed by a verb
phrase, where a verb phrase can be either a verb or a verb phrase ex-
tended by a prepositional phrase:52
(define (parse-sentence)
  (list 'sentence (parse-noun-phrase) (parse-verb-phrase)))
(define (parse-verb-phrase)
  (define (maybe-extend verb-phrase)
     (amb verb-phrase

   52 Observe that this definition is recursive—a verb may be followed by any number

of prepositional phrases.

            (list 'verb-phrase
  (maybe-extend (parse-word verbs)))

While we’re at it, we can also elaborate the definition of noun phrases
to permit such things as “a cat in the class.” What we used to call a
noun phrase, we’ll now call a simple noun phrase, and a noun phrase
will now be either a simple noun phrase or a noun phrase extended by
a prepositional phrase:
(define (parse-simple-noun-phrase)
  (list 'simple-noun-phrase
        (parse-word articles)
        (parse-word nouns)))
(define (parse-noun-phrase)
  (define (maybe-extend noun-phrase)
    (amb noun-phrase
            (list 'noun-phrase
  (maybe-extend (parse-simple-noun-phrase)))

Our new grammar lets us parse more complex sentences. For example
(parse '(the student with the cat sleeps in the class))

  (simple-noun-phrase (article the) (noun student))
   (prep with)

   (simple-noun-phrase (article the) (noun cat))))
  (verb sleeps)
   (prep in)
   (simple-noun-phrase (article the) (noun class)))))

Observe that a given input may have more than one legal parse. In the
sentence “e professor lectures to the student with the cat,” it may be
that the professor is lecturing with the cat, or that the student has the
cat. Our nondeterministic program finds both possibilities:
(parse '(the professor lectures to the student with the cat))

 (simple-noun-phrase (article the) (noun professor))
   (verb lectures)
    (prep to)
    (simple-noun-phrase (article the) (noun student))))
   (prep with)
   (simple-noun-phrase (article the) (noun cat)))))

Asking the evaluator to try again yields
 (simple-noun-phrase (article the) (noun professor))
  (verb lectures)
   (prep to)

 (simple-noun-phrase (article the) (noun student))
  (prep with)
  (simple-noun-phrase (article the) (noun cat)))))))

  Exercise 4.45: With the grammar given above, the follow-
  ing sentence can be parsed in five different ways: “e pro-
  fessor lectures to the student in the class with the cat.” Give
  the five parses and explain the differences in shades of mean-
  ing among them.

  Exercise 4.46: e evaluators in Section 4.1 and Section 4.2
  do not determine what order operands are evaluated in. We
  will see that the amb evaluator evaluates them from le to
  right. Explain why our parsing program wouldn’t work if
  the operands were evaluated in some other order.

  Exercise 4.47: Louis Reasoner suggests that, since a verb
  phrase is either a verb or a verb phrase followed by a prepo-
  sitional phrase, it would be much more straightforward to
  define the procedure parse-verb-phrase as follows (and
  similarly for noun phrases):
  (define (parse-verb-phrase)
    (amb (parse-word verbs)
          (list 'verb-phrase

  Does this work? Does the program’s behavior change if we
  interchange the order of expressions in the amb?

       Exercise 4.48: Extend the grammar given above to handle
       more complex sentences. For example, you could extend
       noun phrases and verb phrases to include adjectives and
       adverbs, or you could handle compound sentences.53

       Exercise 4.49: Alyssa P. Hacker is more interested in gen-
       erating interesting sentences than in parsing them. She rea-
       sons that by simply changing the procedure parse-word so
       that it ignores the “input sentence” and instead always suc-
       ceeds and generates an appropriate word, we can use the
       programs we had built for parsing to do generation instead.
       Implement Alyssa’s idea, and show the first half-dozen or
       so sentences generated.54

4.3.3 Implementing the amb Evaluator
e evaluation of an ordinary Scheme expression may return a value,
may never terminate, or may signal an error. In nondeterministic Scheme
the evaluation of an expression may in addition result in the discovery
of a dead end, in which case evaluation must backtrack to a previous
  53 is   kind of grammar can become arbitrarily complex, but it is only a toy as
far as real language understanding is concerned. Real natural-language understand-
ing by computer requires an elaborate mixture of syntactic analysis and interpretation
of meaning. On the other hand, even toy parsers can be useful in supporting flexi-
ble command languages for programs such as information-retrieval systems. Winston
1992 discusses computational approaches to real language understanding and also the
applications of simple grammars to command languages.
   54 Although Alyssa’s idea works just fine (and is surprisingly simple), the sentences

that it generates are a bit boring—they don’t sample the possible sentences of this lan-
guage in a very interesting way. In fact, the grammar is highly recursive in many places,
and Alyssa’s technique “falls into” one of these recursions and gets stuck. See Exercise
4.50 for a way to deal with this.

choice point. e interpretation of nondeterministic Scheme is compli-
cated by this extra case.
    We will construct the amb evaluator for nondeterministic Scheme by
modifying the analyzing evaluator of Section As in the analyz-
ing evaluator, evaluation of an expression is accomplished by calling an
execution procedure produced by analysis of that expression. e dif-
ference between the interpretation of ordinary Scheme and the inter-
pretation of nondeterministic Scheme will be entirely in the execution

Execution procedures and continuations
Recall that the execution procedures for the ordinary evaluator take one
argument: the environment of execution. In contrast, the execution pro-
cedures in the amb evaluator take three arguments: the environment,
and two procedures called continuation procedures. e evaluation of
an expression will finish by calling one of these two continuations: If
the evaluation results in a value, the success continuation is called with
that value; if the evaluation results in the discovery of a dead end, the
failure continuation is called. Constructing and calling appropriate con-
tinuations is the mechanism by which the nondeterministic evaluator
implements backtracking.
    It is the job of the success continuation to receive a value and pro-
ceed with the computation. Along with that value, the success contin-
uation is passed another failure continuation, which is to be called sub-
sequently if the use of that value leads to a dead end.
  55 We  chose to implement the lazy evaluator in Section 4.2 as a modification of the
ordinary metacircular evaluator of Section 4.1.1. In contrast, we will base the amb eval-
uator on the analyzing evaluator of Section 4.1.7, because the execution procedures in
that evaluator provide a convenient framework for implementing backtracking.

     It is the job of the failure continuation to try another branch of the
nondeterministic process. e essence of the nondeterministic language
is in the fact that expressions may represent choices among alternatives.
e evaluation of such an expression must proceed with one of the indi-
cated alternative choices, even though it is not known in advance which
choices will lead to acceptable results. To deal with this, the evaluator
picks one of the alternatives and passes this value to the success con-
tinuation. Together with this value, the evaluator constructs and passes
along a failure continuation that can be called later to choose a different
     A failure is triggered during evaluation (that is, a failure continua-
tion is called) when a user program explicitly rejects the current line of
aack (for example, a call to require may result in execution of (amb),
an expression that always fails—see Section 4.3.1). e failure continu-
ation in hand at that point will cause the most recent choice point to
choose another alternative. If there are no more alternatives to be con-
sidered at that choice point, a failure at an earlier choice point is trig-
gered, and so on. Failure continuations are also invoked by the driver
loop in response to a try-again request, to find another value of the
     In addition, if a side-effect operation (such as assignment to a vari-
able) occurs on a branch of the process resulting from a choice, it may be
necessary, when the process finds a dead end, to undo the side effect be-
fore making a new choice. is is accomplished by having the side-effect
operation produce a failure continuation that undoes the side effect and
propagates the failure.
     In summary, failure continuations are constructed by

    • amb expressions—to provide a mechanism to make alternative choices
      if the current choice made by the amb expression leads to a dead
    • the top-level driver—to provide a mechanism to report failure
      when the choices are exhausted;

    • assignments—to intercept failures and undo assignments during

Failures are initiated only when a dead end is encountered. is occurs

    • if the user program executes (amb);

    • if the user types try-again at the top-level driver.

Failure continuations are also called during processing of a failure:

    • When the failure continuation created by an assignment finishes
      undoing a side effect, it calls the failure continuation it inter-
      cepted, in order to propagate the failure back to the choice point
      that led to this assignment or to the top level.

    • When the failure continuation for an amb runs out of choices, it
      calls the failure continuation that was originally given to the amb,
      in order to propagate the failure back to the previous choice point
      or to the top level.

Structure of the evaluator
e syntax- and data-representation procedures for the amb evaluator,
and also the basic analyze procedure, are identical to those in the eval-
uator of Section 4.1.7, except for the fact that we need additional syntax
procedures to recognize the amb special form:56
   56 We assume that the evaluator supports let (see Exercise 4.22), which we have used

in our nondeterministic programs.

(define (amb? exp) (tagged-list? exp 'amb))
(define (amb-choices exp) (cdr exp))

We must also add to the dispatch in analyze a clause that will recognize
this special form and generate an appropriate execution procedure:
((amb? exp) (analyze-amb exp))

e top-level procedure ambeval (similar to the version of eval given
in Section 4.1.7) analyzes the given expression and applies the resulting
execution procedure to the given environment, together with two given
(define (ambeval exp env succeed fail)
  ((analyze exp) env succeed fail))

A success continuation is a procedure of two arguments: the value just
obtained and another failure continuation to be used if that value leads
to a subsequent failure. A failure continuation is a procedure of no ar-
guments. So the general form of an execution procedure is
(lambda (env succeed fail)
  ;; succeed is (lambda (value fail) . . .)
  ;; fail is (lambda () . . .)
  . . .)

For example, executing
(ambeval   ⟨exp⟩
           (lambda (value fail) value)
           (lambda () 'failed))

will aempt to evaluate the given expression and will return either the
expression’s value (if the evaluation succeeds) or the symbol failed (if
the evaluation fails). e call to ambeval in the driver loop shown below

uses much more complicated continuation procedures, which continue
the loop and support the try-again request.
    Most of the complexity of the amb evaluator results from the me-
chanics of passing the continuations around as the execution proce-
dures call each other. In going through the following code, you should
compare each of the execution procedures with the corresponding pro-
cedure for the ordinary evaluator given in Section 4.1.7.

Simple expressions
e execution procedures for the simplest kinds of expressions are es-
sentially the same as those for the ordinary evaluator, except for the
need to manage the continuations. e execution procedures simply
succeed with the value of the expression, passing along the failure con-
tinuation that was passed to them.
(define (analyze-self-evaluating exp)
  (lambda (env succeed fail)
    (succeed exp fail)))
(define (analyze-quoted exp)
  (let ((qval (text-of-quotation exp)))
    (lambda (env succeed fail)
      (succeed qval fail))))
(define (analyze-variable exp)
  (lambda (env succeed fail)
    (succeed (lookup-variable-value exp env) fail)))
(define (analyze-lambda exp)
  (let ((vars (lambda-parameters exp))
        (bproc (analyze-sequence (lambda-body exp))))
    (lambda (env succeed fail)
      (succeed (make-procedure vars bproc env) fail))))

Notice that looking up a variable always ‘succeeds.’ If lookup-variable-
value fails to find the variable, it signals an error, as usual. Such a “fail-
ure” indicates a program bug—a reference to an unbound variable; it is
not an indication that we should try another nondeterministic choice
instead of the one that is currently being tried.

Conditionals and sequences
Conditionals are also handled in a similar way as in the ordinary eval-
uator. e execution procedure generated by analyze-if invokes the
predicate execution procedure pproc with a success continuation that
checks whether the predicate value is true and goes on to execute ei-
ther the consequent or the alternative. If the execution of pproc fails,
the original failure continuation for the if expression is called.
(define (analyze-if exp)
  (let ((pproc (analyze (if-predicate exp)))
         (cproc (analyze (if-consequent exp)))
         (aproc (analyze (if-alternative exp))))
    (lambda (env succeed fail)
       (pproc env
               ;; success continuation for evaluating the predicate
               ;; to obtain pred-value
               (lambda (pred-value fail2)
                 (if (true? pred-value)
                      (cproc env succeed fail2)
                      (aproc env succeed fail2)))
               ;; failure continuation for evaluating the predicate

Sequences are also handled in the same way as in the previous evaluator,
except for the machinations in the subprocedure sequentially that are
required for passing the continuations. Namely, to sequentially execute

a   and then b, we call a with a success continuation that calls b.
(define (analyze-sequence exps)
    (define (sequentially a b)
      (lambda (env succeed fail)
        (a env
            ;; success continuation for calling a
            (lambda (a-value fail2)
              (b env succeed fail2))
            ;; failure continuation for calling a
    (define (loop first-proc rest-procs)
      (if (null? rest-procs)
           (loop (sequentially first-proc
                                    (car rest-procs))
                  (cdr rest-procs))))
    (let ((procs (map analyze exps)))
      (if (null? procs)
           (error "Empty sequence: ANALYZE"))
      (loop (car procs) (cdr procs))))

Definitions and assignments
Definitions are another case where we must go to some trouble to man-
age the continuations, because it is necessary to evaluate the definition-
value expression before actually defining the new variable. To accom-
plish this, the definition-value execution procedure vproc is called with
the environment, a success continuation, and the failure continuation.
If the execution of vproc succeeds, obtaining a value val for the defined
variable, the variable is defined and the success is propagated:
(define (analyze-definition exp)

  (let ((var (definition-variable exp))
          (vproc (analyze (definition-value exp))))
     (lambda (env succeed fail)
       (vproc env
                 (lambda (val fail2)
                   (define-variable! var val env)
                   (succeed 'ok fail2))

Assignments are more interesting. is is the first place where we really
use the continuations, rather than just passing them around. e exe-
cution procedure for assignments starts out like the one for definitions.
It first aempts to obtain the new value to be assigned to the variable.
If this evaluation of vproc fails, the assignment fails.
     If vproc succeeds, however, and we go on to make the assignment,
we must consider the possibility that this branch of the computation
might later fail, which will require us to backtrack out of the assign-
ment. us, we must arrange to undo the assignment as part of the
backtracking process.57
     is is accomplished by giving vproc a success continuation (marked
with the comment “*1*” below) that saves the old value of the variable
before assigning the new value to the variable and proceeding from the
assignment. e failure continuation that is passed along with the value
of the assignment (marked with the comment “*2*” below) restores the
old value of the variable before continuing the failure. at is, a suc-
cessful assignment provides a failure continuation that will intercept a
subsequent failure; whatever failure would otherwise have called fail2
calls this procedure instead, to undo the assignment before actually call-
ing fail2.
   57 We didn’t worry about undoing definitions, since we can assume that internal def-

initions are scanned out (Section 4.1.6).

(define (analyze-assignment exp)
  (let ((var (assignment-variable exp))
        (vproc (analyze (assignment-value exp))))
    (lambda (env succeed fail)
      (vproc env
              (lambda (val fail2)            ; *1*
                (let ((old-value
                         (lookup-variable-value var env)))
                   (set-variable-value! var val env)
                   (succeed 'ok
                             (lambda ()      ; *2*
                                  var old-value env)

Procedure applications
e execution procedure for applications contains no new ideas except
for the technical complexity of managing the continuations. is com-
plexity arises in analyze-application, due to the need to keep track of
the success and failure continuations as we evaluate the operands. We
use a procedure get-args to evaluate the list of operands, rather than
a simple map as in the ordinary evaluator.
(define (analyze-application exp)
  (let ((fproc (analyze (operator exp)))
        (aprocs (map analyze (operands exp))))
    (lambda (env succeed fail)
      (fproc env
              (lambda (proc fail2)
                (get-args aprocs

                             (lambda (args fail3)
                                proc args succeed fail3))

In get-args, notice how cdr-ing down the list of aproc execution pro-
cedures and consing up the resulting list of args is accomplished by
calling each aproc in the list with a success continuation that recur-
sively calls get-args. Each of these recursive calls to get-args has a
success continuation whose value is the cons of the newly obtained ar-
gument onto the list of accumulated arguments:
(define (get-args aprocs env succeed fail)
  (if (null? aprocs)
      (succeed '() fail)
      ((car aprocs)
       ;; success continuation for this aproc
       (lambda (arg fail2)
          (cdr aprocs)
          ;; success continuation for
          ;; recursive call to get-args
          (lambda (args fail3)
             (succeed (cons arg args) fail3))

e actual procedure application, which is performed by execute-appli-
cation, is accomplished in the same way as for the ordinary evaluator,
except for the need to manage the continuations.

(define (execute-application proc args succeed fail)
  (cond ((primitive-procedure? proc)
          (succeed (apply-primitive-procedure proc args)
         ((compound-procedure? proc)
          ((procedure-body proc)
            (procedure-parameters proc)
            (procedure-environment proc))
          (error "Unknown procedure type: EXECUTE-APPLICATION"

Evaluating amb expressions
e amb special form is the key element in the nondeterministic lan-
guage. Here we see the essence of the interpretation process and the
reason for keeping track of the continuations. e execution procedure
for amb defines a loop try-next that cycles through the execution pro-
cedures for all the possible values of the amb expression. Each execution
procedure is called with a failure continuation that will try the next one.
When there are no more alternatives to try, the entire amb expression
(define (analyze-amb exp)
  (let ((cprocs (map analyze (amb-choices exp))))
    (lambda (env succeed fail)
      (define (try-next choices)
         (if (null? choices)

              ((car choices)
              (lambda () (try-next (cdr choices))))))
      (try-next cprocs))))

Driver loop
e driver loop for the amb evaluator is complex, due to the mecha-
nism that permits the user to try again in evaluating an expression. e
driver uses a procedure called internal-loop, which takes as argument
a procedure try-again. e intent is that calling try-again should go
on to the next untried alternative in the nondeterministic evaluation.
internal-loop either calls try-again in response to the user typing
try-again at the driver loop, or else starts a new evaluation by calling
    e failure continuation for this call to ambeval informs the user
that there are no more values and re-invokes the driver loop.
    e success continuation for the call to ambeval is more subtle.
We print the obtained value and then invoke the internal loop again
with a try-again procedure that will be able to try the next alterna-
tive. is next-alternative procedure is the second argument that was
passed to the success continuation. Ordinarily, we think of this second
argument as a failure continuation to be used if the current evaluation
branch later fails. In this case, however, we have completed a successful
evaluation, so we can invoke the “failure” alternative branch in order to
search for additional successful evaluations.
(define input-prompt    ";;; Amb-Eval input:")
(define output-prompt ";;; Amb-Eval value:")

(define (driver-loop)
  (define (internal-loop try-again)
    (prompt-for-input input-prompt)
    (let ((input (read)))
      (if (eq? input 'try-again)
             (newline) (display ";;; Starting a new problem ")
              ;; ambeval success
              (lambda (val next-alternative)
                (announce-output output-prompt)
                (user-print val)
                (internal-loop next-alternative))
              ;; ambeval failure
              (lambda ()
                    ";;; There are no more values of")
                (user-print input)
   (lambda ()
     (newline) (display ";;; There is no current problem")

e initial call to internal-loop uses a try-again procedure that com-
plains that there is no current problem and restarts the driver loop. is
is the behavior that will happen if the user types try-again when there
is no evaluation in progress.

      Exercise 4.50: Implement a new special form ramb that is

like amb except that it searches alternatives in a random or-
der, rather than from le to right. Show how this can help
with Alyssa’s problem in Exercise 4.49.

Exercise 4.51: Implement a new kind of assignment called
permanent-set! that is not undone upon failure. For ex-
ample, we can choose two distinct elements from a list and
count the number of trials required to make a successful
choice as follows:
(define count 0)
(let ((x (an-element-of '(a b c)))
      (y (an-element-of '(a b c))))
  (permanent-set! count (+ count 1))
  (require (not (eq? x y)))
  (list x y count))
;;; Starting a new problem
;;; Amb-Eval value:
(a b 2)
;;; Amb-Eval input:
;;; Amb-Eval value:
(a c 3)

What values would have been displayed if we had used
set! here rather than permanent-set! ?

Exercise 4.52: Implement a new construct called if-fail
that permits the user to catch the failure of an expression.
if-fail takes two expressions. It evaluates the first expres-
sion as usual and returns as usual if the evaluation suc-
ceeds. If the evaluation fails, however, the value of the sec-
ond expression is returned, as in the following example:

;;; Amb-Eval input:
(if-fail (let ((x (an-element-of '(1 3 5))))
               (require (even? x))
;;; Starting a new problem
;;; Amb-Eval value:

;;; Amb-Eval input:
(if-fail (let ((x (an-element-of '(1 3 5 8))))
               (require (even? x))
;;; Starting a new problem
;;; Amb-Eval value:

Exercise 4.53: With permanent-set! as described in Exer-
cise 4.51 and if-fail as in Exercise 4.52, what will be the
result of evaluating
(let ((pairs '()))
    (let ((p (prime-sum-pair '(1 3 5 8)
                               '(20 35 110))))
      (permanent-set! pairs (cons p pairs))

Exercise 4.54: If we had not realized that require could be
implemented as an ordinary procedure that uses amb, to be
defined by the user as part of a nondeterministic program,

      we would have had to implement it as a special form. is
      would require syntax procedures
      (define (require? exp)
        (tagged-list? exp 'require))
      (define (require-predicate exp)
        (cadr exp))

      and a new clause in the dispatch in analyze
      ((require? exp) (analyze-require exp))

      as well the procedure analyze-require that handles require
      expressions. Complete the following definition of analyze-

      (define (analyze-require exp)
        (let ((pproc (analyze (require-predicate exp))))
          (lambda (env succeed fail)
            (pproc env
                    (lambda (pred-value fail2)
                      (if   ⟨??⟩
                            (succeed 'ok fail2)))

4.4 Logic Programming
In Chapter 1 we stressed that computer science deals with imperative
(how to) knowledge, whereas mathematics deals with declarative (what
is) knowledge. Indeed, programming languages require that the pro-
grammer express knowledge in a form that indicates the step-by-step
methods for solving particular problems. On the other hand, high-level

languages provide, as part of the language implementation, a substantial
amount of methodological knowledge that frees the user from concern
with numerous details of how a specified computation will progress.
    Most programming languages, including Lisp, are organized around
computing the values of mathematical functions. Expression-oriented
languages (such as Lisp, Fortran, and Algol) capitalize on the “pun” that
an expression that describes the value of a function may also be inter-
preted as a means of computing that value. Because of this, most pro-
gramming languages are strongly biased toward unidirectional compu-
tations (computations with well-defined inputs and outputs). ere are,
however, radically different programming languages that relax this bias.
We saw one such example in Section 3.3.5, where the objects of compu-
tation were arithmetic constraints. In a constraint system the direction
and the order of computation are not so well specified; in carrying out a
computation the system must therefore provide more detailed “how to”
knowledge than would be the case with an ordinary arithmetic compu-
tation. is does not mean, however, that the user is released altogether
from the responsibility of providing imperative knowledge. ere are
many constraint networks that implement the same set of constraints,
and the user must choose from the set of mathematically equivalent
networks a suitable network to specify a particular computation.
    e nondeterministic program evaluator of Section 4.3 also moves
away from the view that programming is about constructing algorithms
for computing unidirectional functions. In a nondeterministic language,
expressions can have more than one value, and, as a result, the compu-
tation is dealing with relations rather than with single-valued functions.
Logic programming extends this idea by combining a relational vision
of programming with a powerful kind of symbolic paern matching

called unification.58
    is approach, when it works, can be a very powerful way to write
programs. Part of the power comes from the fact that a single “what is”
fact can be used to solve a number of different problems that would have
different “how to” components. As an example, consider the append op-
eration, which takes two lists as arguments and combines their elements
to form a single list. In a procedural language such as Lisp, we could
define append in terms of the basic list constructor cons, as we did in
Section 2.2.1:

   58 Logic programming has grown out of a long history of research in automatic the-

orem proving. Early theorem-proving programs could accomplish very lile, because
they exhaustively searched the space of possible proofs. e major breakthrough that
made such a search plausible was the discovery in the early 1960s of the unification
algorithm and the resolution principle (Robinson 1965). Resolution was used, for exam-
ple, by Green and Raphael (1968) (see also Green 1969) as the basis for a deductive
question-answering system. During most of this period, researchers concentrated on
algorithms that are guaranteed to find a proof if one exists. Such algorithms were dif-
ficult to control and to direct toward a proof. Hewi (1969) recognized the possibility
of merging the control structure of a programming language with the operations of a
logic-manipulation system, leading to the work in automatic search mentioned in Sec-
tion 4.3.1 (Footnote 4.47). At the same time that this was being done, Colmerauer, in
Marseille, was developing rule-based systems for manipulating natural language (see
Colmerauer et al. 1973). He invented a programming language called Prolog for repre-
senting those rules. Kowalski (1973; 1979), in Edinburgh, recognized that execution of
a Prolog program could be interpreted as proving theorems (using a proof technique
called linear Horn-clause resolution). e merging of the last two strands led to the
logic-programming movement. us, in assigning credit for the development of logic
programming, the French can point to Prolog’s genesis at the University of Marseille,
while the British can highlight the work at the University of Edinburgh. According to
people at , logic programming was developed by these groups in an aempt to fig-
ure out what Hewi was talking about in his brilliant but impenetrable Ph.D. thesis.
For a history of logic programming, see Robinson 1983.

(define (append x y)
  (if (null? x) y (cons (car x) (append (cdr x) y))))

is procedure can be regarded as a translation into Lisp of the follow-
ing two rules, the first of which covers the case where the first list is
empty and the second of which handles the case of a nonempty list,
which is a cons of two parts:

    • For any list y, the empty list and y append to form y.

    • For any u, v, y, and z, (cons u v) and y append to form (cons u
      z) if v and y append to form z.59

Using the append procedure, we can answer questions such as

          Find the append of (a b) and (c d).

But the same two rules are also sufficient for answering the following
sorts of questions, which the procedure can’t answer:

          Find a list y that appends with (a b) to produce (a b c d).
          Find all x and y that append to form (a b c d).

In a logic programming language, the programmer writes an append
“procedure” by stating the two rules about append given above. “How
to” knowledge is provided automatically by the interpreter to allow this
  59 To see the correspondence between the rules and the procedure, let x in the pro-
cedure (where x is nonempty) correspond to (cons u v) in the rule. en z in the rule
corresponds to the append of (cdr x) and y.

single pair of rules to be used to answer all three types of questions
about append.60
    Contemporary logic programming languages (including the one we
implement here) have substantial deficiencies, in that their general “how
to” methods can lead them into spurious infinite loops or other unde-
sirable behavior. Logic programming is an active field of research in
computer science.61
    Earlier in this chapter we explored the technology of implementing
interpreters and described the elements that are essential to an inter-
preter for a Lisp-like language (indeed, to an interpreter for any con-
ventional language). Now we will apply these ideas to discuss an in-
terpreter for a logic programming language. We call this language the
query language, because it is very useful for retrieving information from
data bases by formulating queries, or questions, expressed in the lan-
guage. Even though the query language is very different from Lisp, we
  60 is   certainly does not relieve the user of the entire problem of how to compute
the answer. ere are many different mathematically equivalent sets of rules for for-
mulating the append relation, only some of which can be turned into effective devices
for computing in any direction. In addition, sometimes “what is” information gives no
clue “how to” compute an answer. For example, consider the problem of computing the
y such that y 2 = x .
   61 Interest in logic programming peaked during the early 80s when the Japanese gov-

ernment began an ambitious project aimed at building superfast computers optimized
to run logic programming languages. e speed of such computers was to be measured
in LIPS (Logical Inferences Per Second) rather than the usual FLOPS (FLoating-point
Operations Per Second). Although the project succeeded in developing hardware and
soware as originally planned, the international computer industry moved in a dif-
ferent direction. See Feigenbaum and Shrobe 1993 for an overview evaluation of the
Japanese project. e logic programming community has also moved on to consider
relational programming based on techniques other than simple paern matching, such
as the ability to deal with numerical constraints such as the ones illustrated in the
constraint-propagation system of Section 3.3.5.

will find it convenient to describe the language in terms of the same gen-
eral framework we have been using all along: as a collection of primitive
elements, together with means of combination that enable us to com-
bine simple elements to create more complex elements and means of ab-
straction that enable us to regard complex elements as single conceptual
units. An interpreter for a logic programming language is considerably
more complex than an interpreter for a language like Lisp. Neverthe-
less, we will see that our query-language interpreter contains many of
the same elements found in the interpreter of Section 4.1. In particu-
lar, there will be an “eval” part that classifies expressions according to
type and an “apply” part that implements the language’s abstraction
mechanism (procedures in the case of Lisp, and rules in the case of logic
programming). Also, a central role is played in the implementation by
a frame data structure, which determines the correspondence between
symbols and their associated values. One additional interesting aspect
of our query-language implementation is that we make substantial use
of streams, which were introduced in Chapter 3.

4.4.1 Deductive Information Retrieval
Logic programming excels in providing interfaces to data bases for in-
formation retrieval. e query language we shall implement in this chap-
ter is designed to be used in this way.
     In order to illustrate what the query system does, we will show how
it can be used to manage the data base of personnel records for Mi-
crosha, a thriving high-technology company in the Boston area. e
language provides paern-directed access to personnel information and
can also take advantage of general rules in order to make logical deduc-

A sample data base
e personnel data base for Microsha contains assertions about com-
pany personnel. Here is the information about Ben Bitdiddle, the resi-
dent computer wizard:
(address (Bitdiddle Ben) (Slumerville (Ridge Road) 10))
(job (Bitdiddle Ben) (computer wizard))
(salary (Bitdiddle Ben) 60000)

Each assertion is a list (in this case a triple) whose elements can them-
selves be lists.
     As resident wizard, Ben is in charge of the company’s computer
division, and he supervises two programmers and one technician. Here
is the information about them:
(address (Hacker Alyssa P) (Cambridge (Mass Ave) 78))
(job (Hacker Alyssa P) (computer programmer))
(salary (Hacker Alyssa P) 40000)
(supervisor (Hacker Alyssa P) (Bitdiddle Ben))

(address (Fect Cy D) (Cambridge (Ames Street) 3))
(job (Fect Cy D) (computer programmer))
(salary (Fect Cy D) 35000)
(supervisor (Fect Cy D) (Bitdiddle Ben))

(address (Tweakit Lem E) (Boston (Bay State Road) 22))
(job (Tweakit Lem E) (computer technician))
(salary (Tweakit Lem E) 25000)
(supervisor (Tweakit Lem E) (Bitdiddle Ben))

ere is also a programmer trainee, who is supervised by Alyssa:
(address (Reasoner Louis) (Slumerville (Pine Tree Road) 80))
(job (Reasoner Louis) (computer programmer trainee))

(salary (Reasoner Louis) 30000)
(supervisor (Reasoner Louis) (Hacker Alyssa P))

All of these people are in the computer division, as indicated by the
word computer as the first item in their job descriptions.
    Ben is a high-level employee. His supervisor is the company’s big
wheel himself:
(supervisor (Bitdiddle Ben) (Warbucks Oliver))
(address (Warbucks Oliver) (Swellesley (Top Heap Road)))
(job (Warbucks Oliver) (administration big wheel))
(salary (Warbucks Oliver) 150000)

Besides the computer division supervised by Ben, the company has an
accounting division, consisting of a chief accountant and his assistant:
(address (Scrooge Eben) (Weston (Shady Lane) 10))
(job (Scrooge Eben) (accounting chief accountant))
(salary (Scrooge Eben) 75000)
(supervisor (Scrooge Eben) (Warbucks Oliver))

(address (Cratchet Robert) (Allston (N Harvard Street) 16))
(job (Cratchet Robert) (accounting scrivener))
(salary (Cratchet Robert) 18000)
(supervisor (Cratchet Robert) (Scrooge Eben))

ere is also a secretary for the big wheel:
(address (Aull DeWitt) (Slumerville (Onion Square) 5))
(job (Aull DeWitt) (administration secretary))
(salary (Aull DeWitt) 25000)
(supervisor (Aull DeWitt) (Warbucks Oliver))

e data base also contains assertions about which kinds of jobs can be
done by people holding other kinds of jobs. For instance, a computer

wizard can do the jobs of both a computer programmer and a computer
(can-do-job (computer wizard) (computer programmer))
(can-do-job (computer wizard) (computer technician))

A computer programmer could fill in for a trainee:
(can-do-job (computer programmer)
              (computer programmer trainee))

Also, as is well known,
(can-do-job (administration secretary)
              (administration big wheel))

Simple queries
e query language allows users to retrieve information from the data
base by posing queries in response to the system’s prompt. For example,
to find all computer programmers one can say
;;; Query input:
(job ?x (computer programmer))

e system will respond with the following items:
;;; Query results:
(job (Hacker Alyssa P) (computer programmer))
(job (Fect Cy D) (computer programmer))

e input query specifies that we are looking for entries in the data base
that match a certain paern. In this example, the paern specifies en-
tries consisting of three items, of which the first is the literal symbol job,
the second can be anything, and the third is the literal list (computer
programmer). e “anything” that can be the second item in the match-
ing list is specified by a paern variable, ?x. e general form of a paern

variable is a symbol, taken to be the name of the variable, preceded by
a question mark. We will see below why it is useful to specify names
for paern variables rather than just puing ? into paerns to repre-
sent “anything.” e system responds to a simple query by showing all
entries in the data base that match the specified paern.
    A paern can have more than one variable. For example, the query
(address ?x ?y)

will list all the employees’ addresses.
    A paern can have no variables, in which case the query simply
determines whether that paern is an entry in the data base. If so, there
will be one match; if not, there will be no matches.
    e same paern variable can appear more than once in a query,
specifying that the same “anything” must appear in each position. is
is why variables have names. For example,
(supervisor ?x ?x)

finds all people who supervise themselves (though there are no such
assertions in our sample data base).
    e query
(job ?x (computer ?type))

matches all job entries whose third item is a two-element list whose first
item is computer:
(job (Bitdiddle Ben) (computer wizard))
(job (Hacker Alyssa P) (computer programmer))
(job (Fect Cy D) (computer programmer))
(job (Tweakit Lem E) (computer technician))

is same paern does not match
(job (Reasoner Louis) (computer programmer trainee))

because the third item in the entry is a list of three elements, and the
paern’s third item specifies that there should be two elements. If we
wanted to change the paern so that the third item could be any list
beginning with computer, we could specify62
(job ?x (computer . ?type))

For example,
(computer . ?type)

matches the data
(computer programmer trainee)

with ?type as the list (programmer trainee). It also matches the data
(computer programmer)

with ?type as the list (programmer), and matches the data

with ?type as the empty list ().
    We can describe the query language’s processing of simple queries
as follows:

    • e system finds all assignments to variables in the query paern
      that satisfy the paern—that is, all sets of values for the variables
      such that if the paern variables are instantiated with (replaced
      by) the values, the result is in the data base.

    • e system responds to the query by listing all instantiations of
      the query paern with the variable assignments that satisfy it.
  62 is   uses the doed-tail notation introduced in Exercise 2.20.

Note that if the paern has no variables, the query reduces to a deter-
mination of whether that paern is in the data base. If so, the empty
assignment, which assigns no values to variables, satisfies that paern
for that data base.

      Exercise 4.55: Give simple queries that retrieve the follow-
      ing information from the data base:

        1. all people supervised by Ben Bitdiddle;
        2. the names and jobs of all people in the accounting di-
        3. the names and addresses of all people who live in Slumerville.

Compound queries
Simple queries form the primitive operations of the query language.
In order to form compound operations, the query language provides
means of combination. One thing that makes the query language a logic
programming language is that the means of combination mirror the
means of combination used in forming logical expressions: and, or, and
not. (Here and, or, and not are not the Lisp primitives, but rather oper-
ations built into the query language.)
    We can use and as follows to find the addresses of all the computer
(and (job ?person (computer programmer))
     (address ?person ?where))

e resulting output is
(and (job (Hacker Alyssa P) (computer programmer))
     (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78)))

(and (job (Fect Cy D) (computer programmer))
        (address (Fect Cy D) (Cambridge (Ames Street) 3)))

In general,
(and    ⟨query1 ⟩ ⟨query2 ⟩ . . . ⟨queryn ⟩)

is satisfied by all sets of values for the paern variables that simultane-
ously satisfy ⟨query 1 ⟩ . . . ⟨queryn ⟩.
     As for simple queries, the system processes a compound query by
finding all assignments to the paern variables that satisfy the query,
then displaying instantiations of the query with those values.
     Another means of constructing compound queries is through or.
For example,
(or (supervisor ?x (Bitdiddle Ben))
       (supervisor ?x (Hacker Alyssa P)))

will find all employees supervised by Ben Bitdiddle or Alyssa P. Hacker:
(or (supervisor (Hacker Alyssa P) (Bitdiddle Ben))
       (supervisor (Hacker Alyssa P) (Hacker Alyssa P)))
(or (supervisor (Fect Cy D) (Bitdiddle Ben))
       (supervisor (Fect Cy D) (Hacker Alyssa P)))
(or (supervisor (Tweakit Lem E) (Bitdiddle Ben))
       (supervisor (Tweakit Lem E) (Hacker Alyssa P)))
(or (supervisor (Reasoner Louis) (Bitdiddle Ben))
       (supervisor (Reasoner Louis) (Hacker Alyssa P)))

In general,
(or    ⟨query1 ⟩ ⟨query2 ⟩ . . . ⟨queryn ⟩)

is satisfied by all sets of values for the paern variables that satisfy at
least one of ⟨query 1 ⟩ . . . ⟨queryn ⟩.
    Compound queries can also be formed with not. For example,

(and (supervisor ?x (Bitdiddle Ben))
       (not (job ?x (computer programmer))))

finds all people supervised by Ben Bitdiddle who are not computer pro-
grammers. In general,
(not   ⟨query1 ⟩)

is satisfied by all assignments to the paern variables that do not satisfy
⟨query 1 ⟩.63
     e final combining form is called lisp-value. When lisp-value
is the first element of a paern, it specifies that the next element is a
Lisp predicate to be applied to the rest of the (instantiated) elements as
arguments. In general,
(lisp-value     ⟨predicate⟩ ⟨arg1 ⟩ . . . ⟨argn ⟩)

will be satisfied by assignments to the paern variables for which the
⟨predicate⟩ applied to the instantiated ⟨arg 1 ⟩ . . . ⟨argn ⟩ is true. For ex-
ample, to find all people whose salary is greater than $30,000 we could
(and (salary ?person ?amount) (lisp-value > ?amount 30000))

       Exercise 4.56: Formulate compound queries that retrieve
       the following information:
  63 Actually, this description of not is valid only for simple cases. e real behavior of

not is more complex. We will examine not’s peculiarities in sections Section 4.4.2 and
Section 4.4.3.
  64 lisp-value should be used only to perform an operation not provided in the query

language. In particular, it should not be used to test equality (since that is what the
matching in the query language is designed to do) or inequality (since that can be done
with the same rule shown below).

          a. the names of all people who are supervised by Ben
             Bitdiddle, together with their addresses;
          b. all people whose salary is less than Ben Bitdiddle’s,
             together with their salary and Ben Bitdiddle’s salary;
          c. all people who are supervised by someone who is not
             in the computer division, together with the supervi-
             sor’s name and job.

In addition to primitive queries and compound queries, the query lan-
guage provides means for abstracting queries. ese are given by rules.
e rule
(rule (lives-near ?person-1 ?person-2)
        (and (address ?person-1 (?town . ?rest-1))
              (address ?person-2 (?town . ?rest-2))
              (not (same ?person-1 ?person-2))))

specifies that two people live near each other if they live in the same
town. e final not clause prevents the rule from saying that all peo-
ple live near themselves. e same relation is defined by a very simple
(rule (same ?x ?x))

  65 Notice  that we do not need same in order to make two things be the same: We
just use the same paern variable for each—in effect, we have one thing instead of two
things in the first place. For example, see ?town in the lives-near rule and ?middle-
manager in the wheel rule below. same is useful when we want to force two things to
be different, such as ?person-1 and ?person-2 in the lives-near rule. Although using
the same paern variable in two parts of a query forces the same value to appear in
both places, using different paern variables does not force different values to appear.
(e values assigned to different paern variables may be the same or different.)

e following rule declares that a person is a “wheel” in an organization
if he supervises someone who is in turn a supervisor:
(rule (wheel ?person)
          (and (supervisor ?middle-manager ?person)
               (supervisor ?x ?middle-manager)))

e general form of a rule is
(rule     ⟨conclusion⟩ ⟨body⟩)

where ⟨conclusion⟩ is a paern and ⟨body ⟩ is any query.66 We can think
of a rule as representing a large (even infinite) set of assertions, namely
all instantiations of the rule conclusion with variable assignments that
satisfy the rule body. When we described simple queries (paerns), we
said that an assignment to variables satisfies a paern if the instantiated
paern is in the data base. But the paern needn’t be explicitly in the
data base as an assertion. It can be an implicit assertion implied by a
rule. For example, the query
(lives-near ?x (Bitdiddle Ben))

results in
(lives-near (Reasoner Louis) (Bitdiddle Ben))
(lives-near (Aull DeWitt) (Bitdiddle Ben))

To find all computer programmers who live near Ben Bitdiddle, we can
(and (job ?x (computer programmer))
        (lives-near ?x (Bitdiddle Ben)))

  66 We  will also allow rules without bodies, as in same, and we will interpret such a
rule to mean that the rule conclusion is satisfied by any values of the variables.

As in the case of compound procedures, rules can be used as parts of
other rules (as we saw with the lives-near rule above) or even be de-
fined recursively. For instance, the rule
(rule (outranked-by ?staff-person ?boss)
      (or (supervisor ?staff-person ?boss)
           (and (supervisor ?staff-person ?middle-manager)
                 (outranked-by ?middle-manager ?boss))))

says that a staff person is outranked by a boss in the organization if the
boss is the person’s supervisor or (recursively) if the person’s supervisor
is outranked by the boss.

      Exercise 4.57: Define a rule that says that person 1 can re-
      place person 2 if either person 1 does the same job as person
      2 or someone who does person 1’s job can also do person 2’s
      job, and if person 1 and person 2 are not the same person.
      Using your rule, give queries that find the following:

         a. all people who can replace Cy D. Fect;
         b. all people who can replace someone who is being paid
            more than they are, together with the two salaries.

      Exercise 4.58: Define a rule that says that a person is a “big
      shot” in a division if the person works in the division but
      does not have a supervisor who works in the division.

      Exercise 4.59: Ben Bitdiddle has missed one meeting too
      many. Fearing that his habit of forgeing meetings could
      cost him his job, Ben decides to do something about it. He
      adds all the weekly meetings of the firm to the Microsha
      data base by asserting the following:

(meeting accounting (Monday 9am))
(meeting administration (Monday 10am))
(meeting computer (Wednesday 3pm))
(meeting administration (Friday 1pm))

Each of the above assertions is for a meeting of an entire di-
vision. Ben also adds an entry for the company-wide meet-
ing that spans all the divisions. All of the company’s em-
ployees aend this meeting.
(meeting whole-company (Wednesday 4pm))

  a. On Friday morning, Ben wants to query the data base
     for all the meetings that occur that day. What query
     should he use?
  b. Alyssa P. Hacker is unimpressed. She thinks it would
     be much more useful to be able to ask for her meetings
     by specifying her name. So she designs a rule that says
     that a person’s meetings include all whole-company
     meetings plus all meetings of that person’s division.
     Fill in the body of Alyssa’s rule.
     (rule (meeting-time ?person ?day-and-time)

  c. Alyssa arrives at work on Wednesday morning and
     wonders what meetings she has to aend that day.
     Having defined the above rule, what query should she
     make to find this out?

Exercise 4.60: By giving the query
(lives-near ?person (Hacker Alyssa P))

      Alyssa P. Hacker is able to find people who live near her,
      with whom she can ride to work. On the other hand, when
      she tries to find all pairs of people who live near each other
      by querying
      (lives-near ?person-1 ?person-2)

      she notices that each pair of people who live near each
      other is listed twice; for example,
      (lives-near (Hacker Alyssa P) (Fect Cy D))
      (lives-near (Fect Cy D) (Hacker Alyssa P))

      Why does this happen? Is there a way to find a list of people
      who live near each other, in which each pair appears only
      once? Explain.

Logic as programs
We can regard a rule as a kind of logical implication: If an assignment
of values to paern variables satisfies the body, then it satisfies the con-
clusion. Consequently, we can regard the query language as having the
ability to perform logical deductions based upon the rules. As an exam-
ple, consider the append operation described at the beginning of Section
4.4. As we said, append can be characterized by the following two rules:

    • For any list y, the empty list and y append to form y.

    • For any u, v, y, and z, (cons u v) and y append to form (cons u
      z) if v and y append to form z.

To express this in our query language, we define two rules for a relation
(append-to-form x y z)

which we can interpret to mean “x and y append to form z”:
(rule (append-to-form () ?y ?y))
(rule (append-to-form (?u . ?v) ?y (?u . ?z))
      (append-to-form ?v ?y ?z))

e first rule has no body, which means that the conclusion holds for
any value of ?y. Note how the second rule makes use of doed-tail no-
tation to name the car and cdr of a list.
    Given these two rules, we can formulate queries that compute the
append of two lists:

;;; Query input:
(append-to-form (a b) (c d) ?z)
;;; Query results:
(append-to-form (a b) (c d) (a b c d))

What is more striking, we can use the same rules to ask the question
“Which list, when appended to (a b), yields (a b c d)?” is is done
as follows:
;;; Query input:
(append-to-form (a b) ?y (a b c d))
;;; Query results:
(append-to-form (a b) (c d) (a b c d))

We can also ask for all pairs of lists that append to form (a b c d):
;;; Query input:
(append-to-form ?x ?y (a b c d))
;;; Query results:
(append-to-form () (a b c d) (a b c d))
(append-to-form (a) (b c d) (a b c d))
(append-to-form (a b) (c d) (a b c d))
(append-to-form (a b c) (d) (a b c d))
(append-to-form (a b c d) () (a b c d))

e query system may seem to exhibit quite a bit of intelligence in using
the rules to deduce the answers to the queries above. Actually, as we
will see in the next section, the system is following a well-determined
algorithm in unraveling the rules. Unfortunately, although the system
works impressively in the append case, the general methods may break
down in more complex cases, as we will see in Section 4.4.3.

      Exercise 4.61: e following rules implement a next-to
      relation that finds adjacent elements of a list:
      (rule (?x next-to ?y in (?x ?y . ?u)))
      (rule (?x next-to ?y in (?v . ?z))
            (?x next-to ?y in ?z))

      What will the response be to the following queries?
      (?x next-to ?y in (1 (2 3) 4))
      (?x next-to   1 in (2 1 3 1))

      Exercise 4.62: Define rules to implement the last-pair
      operation of Exercise 2.17, which returns a list containing
      the last element of a nonempty list. Check your rules on
      queries such as (last-pair (3) ?x), (last-pair (1 2
      3) ?x) and (last-pair (2 ?x) (3)). Do your rules work
      correctly on queries such as (last-pair ?x (3)) ?

      Exercise 4.63: e following data base (see Genesis 4) traces
      the genealogy of the descendants of Ada back to Adam, by
      way of Cain:
      (son Adam Cain)
      (son Cain Enoch)
      (son Enoch Irad)

      (son Irad Mehujael)
      (son Mehujael Methushael)
      (son Methushael Lamech)
      (wife Lamech Ada)
      (son Ada Jabal)
      (son Ada Jubal)

      Formulate rules such as “If S is the son of f , and f is the
      son of G, then S is the grandson of G” and “If W is the wife
      of M , and S is the son of W , then S is the son of M ” (which
      was supposedly more true in biblical times than today) that
      will enable the query system to find the grandson of Cain;
      the sons of Lamech; the grandsons of Methushael. (See Ex-
      ercise 4.69 for some rules to deduce more complicated re-

4.4.2 How the ery System Works
In Section 4.4.4 we will present an implementation of the query inter-
preter as a collection of procedures. In this section we give an overview
that explains the general structure of the system independent of low-
level implementation details. Aer describing the implementation of the
interpreter, we will be in a position to understand some of its limitations
and some of the subtle ways in which the query language’s logical op-
erations differ from the operations of mathematical logic.
    It should be apparent that the query evaluator must perform some
kind of search in order to match queries against facts and rules in the
data base. One way to do this would be to implement the query system
as a nondeterministic program, using the amb evaluator of Section 4.3
(see Exercise 4.78). Another possibility is to manage the search with the
aid of streams. Our implementation follows this second approach.

    e query system is organized around two central operations called
paern matching and unification. We first describe paern matching and
explain how this operation, together with the organization of informa-
tion in terms of streams of frames, enables us to implement both simple
and compound queries. We next discuss unification, a generalization of
paern matching needed to implement rules. Finally, we show how the
entire query interpreter fits together through a procedure that classifies
expressions in a manner analogous to the way eval classifies expres-
sions for the interpreter described in Section 4.1.

Paern matching
A paern matcher is a program that tests whether some datum fits a
specified paern. For example, the data list ((a b) c (a b)) matches
the paern (?x c ?x) with the paern variable ?x bound to (a b).
e same data list matches the paern (?x ?y ?z) with ?x and ?z both
bound to (a b) and ?y bound to c. It also matches the paern ((?x ?y)
c (?x ?y)) with ?x bound to a and ?y bound to b. However, it does not
match the paern (?x a ?y), since that paern specifies a list whose
second element is the symbol a.
    e paern matcher used by the query system takes as inputs a
paern, a datum, and a frame that specifies bindings for various paern
variables. It checks whether the datum matches the paern in a way that
is consistent with the bindings already in the frame. If so, it returns the
given frame augmented by any bindings that may have been determined
by the match. Otherwise, it indicates that the match has failed.
    For example, using the paern (?x ?y ?x) to match (a b a) given
an empty frame will return a frame specifying that ?x is bound to a
and ?y is bound to b. Trying the match with the same paern, the same
datum, and a frame specifying that ?y is bound to a will fail. Trying the

match with the same paern, the same datum, and a frame in which ?y
is bound to b and ?x is unbound will return the given frame augmented
by a binding of ?x to a.
    e paern matcher is all the mechanism that is needed to pro-
cess simple queries that don’t involve rules. For instance, to process the
(job ?x (computer programmer))

we scan through all assertions in the data base and select those that
match the paern with respect to an initially empty frame. For each
match we find, we use the frame returned by the match to instantiate
the paern with a value for ?x.

Streams of frames
e testing of paerns against frames is organized through the use of
streams. Given a single frame, the matching process runs through the
data-base entries one by one. For each data-base entry, the matcher gen-
erates either a special symbol indicating that the match has failed or an
extension to the frame. e results for all the data-base entries are col-
lected into a stream, which is passed through a filter to weed out the
failures. e result is a stream of all the frames that extend the given
frame via a match to some assertion in the data base.67
  67 Because  matching is generally very expensive, we would like to avoid applying
the full matcher to every element of the data base. is is usually arranged by breaking
up the process into a fast, coarse match and the final match. e coarse match filters
the data base to produce a small set of candidates for the final match. With care, we
can arrange our data base so that some of the work of coarse matching can be done
when the data base is constructed rather then when we want to select the candidates.
is is called indexing the data base. ere is a vast technology built around data-base-
indexing schemes. Our implementation, described in Section 4.4.4, contains a simple-
minded form of such an optimization.

       input stream                          output stream of frames,
       of frames            query
                                             filtered and extended

                         (job ?x ?y)

                      stream of assertions
                         from data base

          Figure 4.4: A query processes a stream of frames.

    In our system, a query takes an input stream of frames and per-
forms the above matching operation for every frame in the stream, as
indicated in Figure 4.4. at is, for each frame in the input stream, the
query generates a new stream consisting of all extensions to that frame
by matches to assertions in the data base. All these streams are then
combined to form one huge stream, which contains all possible exten-
sions of every frame in the input stream. is stream is the output of
the query.
    To answer a simple query, we use the query with an input stream
consisting of a single empty frame. e resulting output stream contains
all extensions to the empty frame (that is, all answers to our query).
is stream of frames is then used to generate a stream of copies of the
original query paern with the variables instantiated by the values in
each frame, and this is the stream that is finally printed.

Compound queries
e real elegance of the stream-of-frames implementation is evident
when we deal with compound queries. e processing of compound

       input stream          (and A B)           output stream
       of frames                                 of frames
                            A               B

                                data base

      Figure 4.5: e and combination of two queries is produced
      by operating on the stream of frames in series.

queries makes use of the ability of our matcher to demand that a match
be consistent with a specified frame. For example, to handle the and of
two queries, such as
(and (can-do-job ?x (computer programmer trainee))
     (job ?person ?x))

(informally, “Find all people who can do the job of a computer program-
mer trainee”), we first find all entries that match the paern
(can-do-job ?x (computer programmer trainee))

is produces a stream of frames, each of which contains a binding for
?x. en for each frame in the stream we find all entries that match

(job ?person ?x)

in a way that is consistent with the given binding for ?x. Each such
match will produce a frame containing bindings for ?x and ?person.
e and of two queries can be viewed as a series combination of the
two component queries, as shown in Figure 4.5. e frames that pass

                              (or A B)

  input stream                                       output stream
  of frames                                          of frames


                         data base

      Figure 4.6: e or combination of two queries is produced
      by operating on the stream of frames in parallel and merg-
      ing the results.

through the first query filter are filtered and further extended by the
second query.
    Figure 4.6 shows the analogous method for computing the or of two
queries as a parallel combination of the two component queries. e
input stream of frames is extended separately by each query. e two
resulting streams are then merged to produce the final output stream.
    Even from this high-level description, it is apparent that the pro-
cessing of compound queries can be slow. For example, since a query
may produce more than one output frame for each input frame, and
each query in an and gets its input frames from the previous query, an
and query could, in the worst case, have to perform a number of matches

that is exponential in the number of queries (see Exercise 4.76).68 ough
systems for handling only simple queries are quite practical, dealing
with complex queries is extremely difficult.69
    From the stream-of-frames viewpoint, the not of some query acts
as a filter that removes all frames for which the query can be satisfied.
For instance, given the paern
(not (job ?x (computer programmer)))

we aempt, for each frame in the input stream, to produce extension
frames that satisfy (job ?x (computer programmer)). We remove
from the input stream all frames for which such extensions exist. e
result is a stream consisting of only those frames in which the binding
for ?x does not satisfy (job ?x (computer programmer)). For example,
in processing the query
(and (supervisor ?x ?y)
      (not (job ?x (computer programmer))))

the first clause will generate frames with bindings for ?x and ?y. e not
clause will then filter these by removing all frames in which the binding
for ?x satisfies the restriction that ?x is a computer programmer.70
    e lisp-value special form is implemented as a similar filter on
frame streams. We use each frame in the stream to instantiate any vari-
ables in the paern, then apply the Lisp predicate. We remove from the
input stream all frames for which the predicate fails.
  68 But this kind of exponential explosion is not common in and queries because the

added conditions tend to reduce rather than expand the number of frames produced.
  69 ere is a large literature on data-base-management systems that is concerned with

how to handle complex queries efficiently.
  70 ere is a subtle difference between this filter implementation of not and the usual

meaning of not in mathematical logic. See Section 4.4.3.

In order to handle rules in the query language, we must be able to find
the rules whose conclusions match a given query paern. Rule conclu-
sions are like assertions except that they can contain variables, so we
will need a generalization of paern matching—called unification—in
which both the “paern” and the “datum” may contain variables.
    A unifier takes two paerns, each containing constants and vari-
ables, and determines whether it is possible to assign values to the vari-
ables that will make the two paerns equal. If so, it returns a frame
containing these bindings. For example, unifying (?x a ?y) and (?y
?z a) will specify a frame in which ?x, ?y, and ?z must all be bound
to a. On the other hand, unifying (?x ?y a) and (?x b ?y) will fail,
because there is no value for ?y that can make the two paerns equal.
(For the second elements of the paerns to be equal, ?y would have to
be b; however, for the third elements to be equal, ?y would have to be
a.) e unifier used in the query system, like the paern matcher, takes
a frame as input and performs unifications that are consistent with this
    e unification algorithm is the most technically difficult part of the
query system. With complex paerns, performing unification may seem
to require deduction. To unify (?x ?x) and ((a ?y c) (a b ?z)), for
example, the algorithm must infer that ?x should be (a b c), ?y should
be b, and ?z should be c. We may think of this process as solving a
set of equations among the paern components. In general, these are
simultaneous equations, which may require substantial manipulation
to solve.71 For example, unifying (?x ?x) and ((a ?y c) (a b ?z))
may be thought of as specifying the simultaneous equations
  71 In one-sided paern matching, all the equations that contain paern variables are

explicit and already solved for the unknown (the paern variable).

?x   =   (a ?y c)
?x   =   (a b ?z)

ese equations imply that
(a ?y c)       =   (a b ?z)

which in turn implies that
 a   =   a,
?y   =   b,
 c   =   ?z,

and hence that
?x   =   (a b c)

In a successful paern match, all paern variables become bound, and
the values to which they are bound contain only constants. is is also
true of all the examples of unification we have seen so far. In general,
however, a successful unification may not completely determine the
variable values; some variables may remain unbound and others may
be bound to values that contain variables.
    Consider the unification of (?x a) and ((b ?y) ?z). We can deduce
that ?x = (b ?y) and a = ?z, but we cannot further solve for ?x or
?y. e unification doesn’t fail, since it is certainly possible to make the
two paerns equal by assigning values to ?x and ?y. Since this match
in no way restricts the values ?y can take on, no binding for ?y is put
into the result frame. e match does, however, restrict the value of ?x.
Whatever value ?y has, ?x must be (b ?y). A binding of ?x to the paern
(b ?y) is thus put into the frame. If a value for ?y is later determined and
added to the frame (by a paern match or unification that is required
to be consistent with this frame), the previously bound ?x will refer to
this value.72
  72 Another   way to think of unification is that it generates the most general paern

Applying rules
Unification is the key to the component of the query system that makes
inferences from rules. To see how this is accomplished, consider pro-
cessing a query that involves applying a rule, such as
(lives-near ?x (Hacker Alyssa P))

To process this query, we first use the ordinary paern-match procedure
described above to see if there are any assertions in the data base that
match this paern. (ere will not be any in this case, since our data base
includes no direct assertions about who lives near whom.) e next step
is to aempt to unify the query paern with the conclusion of each rule.
We find that the paern unifies with the conclusion of the rule
(rule (lives-near ?person-1 ?person-2)
        (and (address ?person-1 (?town . ?rest-1))
              (address ?person-2 (?town . ?rest-2))
              (not (same ?person-1 ?person-2))))

resulting in a frame specifying that ?person-2 is bound to (Hacker
Alyssa P) and that ?x should be bound to (have the same value as)
?person-1. Now, relative to this frame, we evaluate the compound query
given by the body of the rule. Successful matches will extend this frame
by providing a binding for ?person-1, and consequently a value for ?x,
which we can use to instantiate the original query paern.
    In general, the query evaluator uses the following method to apply
a rule when trying to establish a query paern in a frame that specifies
bindings for some of the paern variables:
that is a specialization of the two input paerns. at is, the unification of (?x a) and
((b ?y) ?z) is ((b ?y) a), and the unification of (?x a ?y) and (?y ?z a), discussed
above, is (a a a). For our implementation, it is more convenient to think of the result
of unification as a frame rather than a paern.

    • Unify the query with the conclusion of the rule to form, if suc-
      cessful, an extension of the original frame.

    • Relative to the extended frame, evaluate the query formed by the
      body of the rule.

Notice how similar this is to the method for applying a procedure in the
eval/apply evaluator for Lisp:

    • Bind the procedure’s parameters to its arguments to form a frame
      that extends the original procedure environment.

    • Relative to the extended environment, evaluate the expression
      formed by the body of the procedure.

e similarity between the two evaluators should come as no surprise.
Just as procedure definitions are the means of abstraction in Lisp, rule
definitions are the means of abstraction in the query language. In each
case, we unwind the abstraction by creating appropriate bindings and
evaluating the rule or procedure body relative to these.

Simple queries
We saw earlier in this section how to evaluate simple queries in the
absence of rules. Now that we have seen how to apply rules, we can
describe how to evaluate simple queries by using both rules and asser-
    Given the query paern and a stream of frames, we produce, for
each frame in the input stream, two streams:

    • a stream of extended frames obtained by matching the paern
      against all assertions in the data base (using the paern matcher),

    • a stream of extended frames obtained by applying all possible
      rules (using the unifier).73

Appending these two streams produces a stream that consists of all the
ways that the given paern can be satisfied consistent with the original
frame. ese streams (one for each frame in the input stream) are now
all combined to form one large stream, which therefore consists of all
the ways that any of the frames in the original input stream can be
extended to produce a match with the given paern.

The query evaluator and the driver loop
Despite the complexity of the underlying matching operations, the sys-
tem is organized much like an evaluator for any language. e proce-
dure that coordinates the matching operations is called qeval, and it
plays a role analogous to that of the eval procedure for Lisp. qeval
takes as inputs a query and a stream of frames. Its output is a stream of
frames, corresponding to successful matches to the query paern, that
extend some frame in the input stream, as indicated in Figure 4.4. Like
eval, qeval classifies the different types of expressions (queries) and
dispatches to an appropriate procedure for each. ere is a procedure
for each special form (and, or, not, and lisp-value) and one for simple
    e driver loop, which is analogous to the driver-loop procedure
for the other evaluators in this chapter, reads queries from the terminal.
For each query, it calls qeval with the query and a stream that consists
  73 Since unification is a generalization of matching, we could simplify the system

by using the unifier to produce both streams. Treating the easy case with the simple
matcher, however, illustrates how matching (as opposed to full-blown unification) can
be useful in its own right.

of a single empty frame. is will produce the stream of all possible
matches (all possible extensions to the empty frame). For each frame in
the resulting stream, it instantiates the original query using the values
of the variables found in the frame. is stream of instantiated queries
is then printed.74
     e driver also checks for the special command assert!, which sig-
nals that the input is not a query but rather an assertion or rule to be
added to the data base. For instance,
(assert! (job (Bitdiddle Ben)
                   (computer wizard)))
(assert! (rule (wheel ?person)
                    (and (supervisor ?middle-manager ?person)
                           (supervisor ?x ?middle-manager))))

4.4.3 Is Logic Programming Mathematical Logic?
e means of combination used in the query language may at first seem
identical to the operations and, or, and not of mathematical logic, and
the application of query-language rules is in fact accomplished through
a legitimate method of inference.75 is identification of the query lan-
guage with mathematical logic is not really valid, though, because the
  74 e  reason we use streams (rather than lists) of frames is that the recursive appli-
cation of rules can generate infinite numbers of values that satisfy a query. e delayed
evaluation embodied in streams is crucial here: e system will print responses one by
one as they are generated, regardless of whether there are a finite or infinite number
of responses.
   75 at a particular method of inference is legitimate is not a trivial assertion. One

must prove that if one starts with true premises, only true conclusions can be derived.
e method of inference represented by rule applications is modus ponens, the familiar
method of inference that says that if A is true and A implies B is true, then we may
conclude that B is true.

query language provides a control structure that interprets the logical
statements procedurally. We can oen take advantage of this control
structure. For example, to find all of the supervisors of programmers
we could formulate a query in either of two logically equivalent forms:
(and (job ?x (computer programmer)) (supervisor ?x ?y))

(and (supervisor ?x ?y) (job ?x (computer programmer)))

If a company has many more supervisors than programmers (the usual
case), it is beer to use the first form rather than the second because the
data base must be scanned for each intermediate result (frame) produced
by the first clause of the and.
     e aim of logic programming is to provide the programmer with
techniques for decomposing a computational problem into two separate
problems: “what” is to be computed, and “how” this should be computed.
is is accomplished by selecting a subset of the statements of mathe-
matical logic that is powerful enough to be able to describe anything
one might want to compute, yet weak enough to have a controllable
procedural interpretation. e intention here is that, on the one hand, a
program specified in a logic programming language should be an effec-
tive program that can be carried out by a computer. Control (“how” to
compute) is effected by using the order of evaluation of the language.
We should be able to arrange the order of clauses and the order of sub-
goals within each clause so that the computation is done in an order
deemed to be effective and efficient. At the same time, we should be
able to view the result of the computation (“what” to compute) as a
simple consequence of the laws of logic.
     Our query language can be regarded as just such a procedurally in-
terpretable subset of mathematical logic. An assertion represents a sim-

ple fact (an atomic proposition). A rule represents the implication that
the rule conclusion holds for those cases where the rule body holds. A
rule has a natural procedural interpretation: To establish the conclusion
of the rule, establish the body of the rule. Rules, therefore, specify com-
putations. However, because rules can also be regarded as statements
of mathematical logic, we can justify any “inference” accomplished by
a logic program by asserting that the same result could be obtained by
working entirely within mathematical logic.76

Infinite loops
A consequence of the procedural interpretation of logic programs is that
it is possible to construct hopelessly inefficient programs for solving
certain problems. An extreme case of inefficiency occurs when the sys-
tem falls into infinite loops in making deductions. As a simple example,
suppose we are seing up a data base of famous marriages, including
(assert! (married Minnie Mickey))

If we now ask
(married Mickey ?who)

   76 We must qualify this statement by agreeing that, in speaking of the “inference”

accomplished by a logic program, we assume that the computation terminates. Unfor-
tunately, even this qualified statement is false for our implementation of the query lan-
guage (and also false for programs in Prolog and most other current logic programming
languages) because of our use of not and lisp-value. As we will describe below, the
not implemented in the query language is not always consistent with the not of math-
ematical logic, and lisp-value introduces additional complications. We could imple-
ment a language consistent with mathematical logic by simply removing not and lisp-
value from the language and agreeing to write programs using only simple queries, and,
and or. However, this would greatly restrict the expressive power of the language. One
of the major concerns of research in logic programming is to find ways to achieve more
consistency with mathematical logic without unduly sacrificing expressive power.

we will get no response, because the system doesn’t know that if A is
married to B, then B is married to A. So we assert the rule
(assert! (rule (married ?x ?y) (married ?y ?x)))

and again query
(married Mickey ?who)

Unfortunately, this will drive the system into an infinite loop, as follows:

    • e system finds that the married rule is applicable; that is, the
      rule conclusion (married ?x ?y) successfully unifies with the
      query paern (married Mickey ?who) to produce a frame in
      which ?x is bound to Mickey and ?y is bound to ?who. So the inter-
      preter proceeds to evaluate the rule body (married ?y ?x) in this
      frame—in effect, to process the query (married ?who Mickey).

    • One answer appears directly as an assertion in the data base:
      (married Minnie Mickey).

    • e married rule is also applicable, so the interpreter again eval-
      uates the rule body, which this time is equivalent to (married
      Mickey ?who).

e system is now in an infinite loop. Indeed, whether the system will
find the simple answer (married Minnie Mickey) before it goes into
the loop depends on implementation details concerning the order in
which the system checks the items in the data base. is is a very sim-
ple example of the kinds of loops that can occur. Collections of interre-
lated rules can lead to loops that are much harder to anticipate, and the
appearance of a loop can depend on the order of clauses in an and (see

Exercise 4.64) or on low-level details concerning the order in which the
system processes queries.77

Problems with not
Another quirk in the query system concerns not. Given the data base
of Section 4.4.1, consider the following two queries:
(and (supervisor ?x ?y)
       (not (job ?x (computer programmer))))
(and (not (job ?x (computer programmer)))
       (supervisor ?x ?y))

ese two queries do not produce the same result. e first query begins
by finding all entries in the data base that match (supervisor ?x ?y),
and then filters the resulting frames by removing the ones in which the
value of ?x satisfies (job ?x (computer programmer)). e second
query begins by filtering the incoming frames to remove those that can
satisfy (job ?x (computer programmer)). Since the only incoming
frame is empty, it checks the data base to see if there are any paerns
that satisfy (job ?x (computer programmer)). Since there generally
   77 is is not a problem of the logic but one of the procedural interpretation of the

logic provided by our interpreter. We could write an interpreter that would not fall
into a loop here. For example, we could enumerate all the proofs derivable from our
assertions and our rules in a breadth-first rather than a depth-first order. However,
such a system makes it more difficult to take advantage of the order of deductions
in our programs. One aempt to build sophisticated control into such a program is
described in deKleer et al. 1977. Another technique, which does not lead to such serious
control problems, is to put in special knowledge, such as detectors for particular kinds of
loops (Exercise 4.67). However, there can be no general scheme for reliably preventing a
system from going down infinite paths in performing deductions. Imagine a diabolical
rule of the form “To show P (x ) is true, show that P (f (x )) is true,” for some suitably
chosen function f .

are entries of this form, the not clause filters out the empty frame and
returns an empty stream of frames. Consequently, the entire compound
query returns an empty stream.
    e trouble is that our implementation of not really is meant to
serve as a filter on values for the variables. If a not clause is processed
with a frame in which some of the variables remain unbound (as does
?x in the example above), the system will produce unexpected results.
Similar problems occur with the use of lisp-value—the Lisp predicate
can’t work if some of its arguments are unbound. See Exercise 4.77.
    ere is also a much more serious way in which the not of the query
language differs from the not of mathematical logic. In logic, we inter-
pret the statement “not P ” to mean that P is not true. In the query sys-
tem, however, “not P ” means that P is not deducible from the knowledge
in the data base. For example, given the personnel data base of Section
4.4.1, the system would happily deduce all sorts of not statements, such
as that Ben Bitdiddle is not a baseball fan, that it is not raining outside,
and that 2 + 2 is not 4.78 In other words, the not of logic programming
languages reflects the so-called closed world assumption that all relevant
information has been included in the data base.79

       Exercise 4.64: Louis Reasoner mistakenly deletes the outranked-
       by rule (Section 4.4.1) from the data base. When he real-
       izes this, he quickly reinstalls it. Unfortunately, he makes a
       slight change in the rule, and types it in as
  78 Consider  the query (not (baseball-fan (Bitdiddle Ben))). e system finds
that (baseball-fan (Bitdiddle Ben)) is not in the data base, so the empty frame
does not satisfy the paern and is not filtered out of the initial stream of frames. e
result of the query is thus the empty frame, which is used to instantiate the input query
to produce (not (baseball-fan (Bitdiddle Ben))).
   79 A discussion and justification of this treatment of not can be found in the article

by Clark (1978).

(rule (outranked-by ?staff-person ?boss)
      (or (supervisor ?staff-person ?boss)
           (and (outranked-by ?middle-manager ?boss)
                 (supervisor ?staff-person

Just aer Louis types this information into the system, De-
Wi Aull comes by to find out who outranks Ben Bitdiddle.
He issues the query
(outranked-by (Bitdiddle Ben) ?who)

Aer answering, the system goes into an infinite loop. Ex-
plain why.

Exercise 4.65: Cy D. Fect, looking forward to the day when
he will rise in the organization, gives a query to find all the
wheels (using the wheel rule of Section 4.4.1):
(wheel ?who)

To his surprise, the system responds
;;; Query results:
(wheel (Warbucks Oliver))
(wheel (Bitdiddle Ben))
(wheel (Warbucks Oliver))
(wheel (Warbucks Oliver))
(wheel (Warbucks Oliver))

Why is Oliver Warbucks listed four times?

Exercise 4.66: Ben has been generalizing the query sys-
tem to provide statistics about the company. For example,

to find the total salaries of all the computer programmers
one will be able to say
(sum ?amount (and (job ?x (computer programmer))
                    (salary ?x ?amount)))

In general, Ben’s new system allows expressions of the form
(accumulation-function    ⟨variable⟩ ⟨query pattern⟩)

where accumulation-function can be things like sum, average,
or maximum. Ben reasons that it should be a cinch to imple-
ment this. He will simply feed the query paern to qeval.
is will produce a stream of frames. He will then pass this
stream through a mapping function that extracts the value
of the designated variable from each frame in the stream
and feed the resulting stream of values to the accumulation
function. Just as Ben completes the implementation and is
about to try it out, Cy walks by, still puzzling over the wheel
query result in Exercise 4.65. When Cy shows Ben the sys-
tem’s response, Ben groans, “Oh, no, my simple accumula-
tion scheme won’t work!”
What has Ben just realized? Outline a method he can use
to salvage the situation.

Exercise 4.67: Devise a way to install a loop detector in the
query system so as to avoid the kinds of simple loops illus-
trated in the text and in Exercise 4.64. e general idea is
that the system should maintain some sort of history of its
current chain of deductions and should not begin process-
ing a query that it is already working on. Describe what
kind of information (paerns and frames) is included in

     this history, and how the check should be made. (Aer you
     study the details of the query-system implementation in
     Section 4.4.4, you may want to modify the system to in-
     clude your loop detector.)

     Exercise 4.68: Define rules to implement the reverse op-
     eration of Exercise 2.18, which returns a list containing the
     same elements as a given list in reverse order. (Hint: Use
     append-to-form.) Can your rules answer both (reverse
     (1 2 3) ?x) and (reverse ?x (1 2 3)) ?

     Exercise 4.69: Beginning with the data base and the rules
     you formulated in Exercise 4.63, devise a rule for adding
     “greats” to a grandson relationship. is should enable the
     system to deduce that Irad is the great-grandson of Adam,
     or that Jabal and Jubal are the great-great-great-great-great-
     grandsons of Adam. (Hint: Represent the fact about Irad, for
     example, as ((great grandson) Adam Irad). Write rules
     that determine if a list ends in the word grandson. Use this
     to express a rule that allows one to derive the relationship
     ((great . ?rel) ?x ?y), where ?rel is a list ending in
     grandson.) Check your rules on queries such as ((great
     grandson) ?g ?ggs) and (?relationship Adam Irad).

4.4.4 Implementing the ery System
Section 4.4.2 described how the query system works. Now we fill in the
details by presenting a complete implementation of the system.

                                  635 The Driver Loop and Instantiation
e driver loop for the query system repeatedly reads input expressions.
If the expression is a rule or assertion to be added to the data base, then
the information is added. Otherwise the expression is assumed to be
a query. e driver passes this query to the evaluator qeval together
with an initial frame stream consisting of a single empty frame. e
result of the evaluation is a stream of frames generated by satisfying
the query with variable values found in the data base. ese frames are
used to form a new stream consisting of copies of the original query in
which the variables are instantiated with values supplied by the stream
of frames, and this final stream is printed at the terminal:
(define input-prompt       ";;; Query input:")
(define output-prompt ";;; Query results:")

(define (query-driver-loop)
  (prompt-for-input input-prompt)
  (let ((q (query-syntax-process (read))))
    (cond ((assertion-to-be-added? q)
            (add-rule-or-assertion! (add-assertion-body q))
            (display "Assertion added to data base.")
            (display output-prompt)
               (lambda (frame)

                  (lambda (v f)